1. Ch. 5: Subgroups & Product Groups!

2. DEFINITION: A group is a set (denoted G) with an algebraic operation (denoted ) that satisfies the following properties: (1) G has an “identity” (denoted I) which has no effect on other members; that is, AI = A and IA = A for all members, A, of G. (2) Each member, A, of G has an inverse in G (usually denoted A –1 ), which combines with it in either order to give the identity: AA –1 = I and A –1 A = I. (3) The associative property holds: (AB)C = A(BC) for all triples A,B,C of members of G. DEFINITION: A group is a set (denoted G) with an algebraic operation (denoted ) that satisfies the following properties: (1) G has an “identity” (denoted I) which has no effect on other members; that is, AI = A and IA = A for all members, A, of G. (2) Each member, A, of G has an inverse in G (usually denoted A –1 ), which combines with it in either order to give the identity: AA –1 = I and A –1 A = I. (3) The associative property holds: (AB)C = A(BC) for all triples A,B,C of members of G. Here are three examples of groups: (1) The numbers (under addition). The identity is 0 The inverse of 35 is -35. (3) The 8 symmetries of a square (under composition). The identity is I The inverse of R 90 is R 270. The inverse of H is H. (2) The numbers (under multiplication). The identity is 1 The inverse of 35 is 1/35.

3. D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } includes: *IR 90 R 180 R 270 IIR 90 R 180 R 270 R 90 R 180 R 270 I R 180 R 270 IR 90 R 270 IR 90 R 180 *HVDD’ HIR 180 R 90 R 270 VR 180 IR 270 R 90 DR 270 R 90 IR 180 D’R 90 R 270 R 180 I { H, V, D, D ’ } “the flips” Which color forms a self-contained group on its own? { I, R 90, R 180, R 270 } “the rotations”

4. D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } includes: *IR 90 R 180 R 270 IIR 90 R 180 R 270 R 90 R 180 R 270 I R 180 R 270 IR 90 R 270 IR 90 R 180 *HVDD’ HIR 180 R 90 R 270 VR 180 IR 270 R 90 DR 270 R 90 IR 180 D’R 90 R 270 R 180 I “The rotations” is a group on its own (called C 4 ); This is a Cayley table “the flips” is NOT a group on its own; This is NOT a Cayley table { H, V, D, D ’ } “the flips” { I, R 90, R 180, R 270 } “the rotations”

5. D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } includes: *IR 90 R 180 R 270 IIR 90 R 180 R 270 R 90 R 180 R 270 I R 180 R 270 IR 90 R 270 IR 90 R 180 *HVDD’ HIR 180 R 90 R 270 VR 180 IR 270 R 90 DR 270 R 90 IR 180 D’R 90 R 270 R 180 I If you color some of the members of a group, under what conditions will the colored collection form a self-contained group on its own? { H, V, D, D ’ } “the flips” { I, R 90, R 180, R 270 } “the rotations”

6. If you list all the members of G, and color some of them red, then the red ones form a subgroup (a self-contained group on its own) if… (1) You colored the identity red. (2) When any pair of red members are combined, the answer is red. (3) The inverse of any red member is red. D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } includes:

7. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. If you list all the members of G, and color some of them red, then the red ones form a subgroup (a self-contained group on its own) if… (1) You colored the identity red. (2) When any pair of red members are combined, the answer is red. (3) The inverse of any red member is red.

8. EXAMPLE: In D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } Do the red members {I, R 180, H, V} form a subgroup? DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H.

9. EXAMPLE: In D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } Do the red members {I, R 180, H, V} form a subgroup? YES! This subgroup is called D 2. Think of it as the symmetry group of the “striped square” DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H.

10. In the symmetry group of the W-border pattern, Do the translations alone form a subgroup? DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. W W W W W W W W

11. In the symmetry group of the W-border pattern, Do the translations alone form a subgroup? YES! DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. W W W W W W W W

12. In the symmetry group of the W-border pattern, Do the translations alone form a subgroup? What about the vertical flips alone? YES! DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. W W W W W W W W

13. In the symmetry group of the W-border pattern, Do the translations alone form a subgroup? What about the vertical flips alone? YES! NO! DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. W W W W W W W W

14. In the additive group of integers, Z = {…,-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, …}, Do the even numbers, E = {…, -8, -6, -4, -2, 0, 2, 4, 6, 8, …} form a subgroup? What about the odd numbers? DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H.

15. In the additive group of integers, Z = {…,-8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, …}, Do the even numbers, E = {…, -8, -6, -4, -2, 0, 2, 4, 6, 8, …} form a subgroup? What about the odd numbers? YES! NO! DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H.

16. In the cyclic group, C 10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup? DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H.

17. In the cyclic group, C 10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup? YES! DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H.

18. In the cyclic group, C 10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup? YES! DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. In the cyclic group, C 9 = {0, 1, 2, 3, 4, 5, 6, 7, 8} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup?

19. In the cyclic group, C 10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup? YES! DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. In the cyclic group, C 9 = {0, 1, 2, 3, 4, 5, 6, 7, 8} do the even numbers, E = {0, 2, 4, 6, 8} form a subgroup? NO!

20. C 12 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup? Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup?

21. C 12 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup? Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup? = “the subgroup of C 12 generated by 2”

22. C 12 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. Oops, your little brother colored the 5 with his red crayon. To fix it, what else must you color red to make it a subgroup? Oops, your little brother colored the 5 with his red crayon. To fix it, what else must you color red to make it a subgroup?

23. C 12 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. Oops, your little brother colored the 5 with his red crayon. To fix it, what else must you color red to make it a subgroup? Oops, your little brother colored the 5 with his red crayon. To fix it, what else must you color red to make it a subgroup? = “the subgroup of C 12 generated by 5”

24. Z = {…-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup? Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup?

25. Z = {…-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup? Oops, your little brother colored the 2 with his red crayon. To fix it, what else must you color red to make it a subgroup? = “the subgroup of Z generated by 2”

26. Z = {…-10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: Suppose that G is a group. A collection, H, of G’s members is called a subgroup if it forms a self-contained group on its own, which means that it satisfies all three of these requirements: (Identity) H must include the identity of G. (Products) If A and B are in H, then AB must be in H. (Inverses) The inverse of anything in H must be in H. DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) = “the subgroup of Z generated by 2”

27. DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } Find the subgroup of D 4 generated by each of its 8 members. = ?

28. DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } Find the subgroup of D 4 generated by each of its 8 members. = { I } = { I, R 90, R 180, R 270 } = { I, R 180 } = { I, R 270, R 180, R 90 } = { I, H } = { I, V } = { I, D } = { I, D’ } the same

29. DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } Find the subgroup of D 4 generated by each of its 8 members. THEOREM: If G is a finite group, and A is a member of G, then = { I, A, AA, AAA, AAAA, …} (this list starts repeating as soon as one of these expressions equals I, and not before). THEOREM: If G is a finite group, and A is a member of G, then = { I, A, AA, AAA, AAAA, …} (this list starts repeating as soon as one of these expressions equals I, and not before). (you don’t need to worry about inverses)

30. DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) DEFINITION: If G is a group, and A is a member of G, then, = {…, A –1 A –1 A –1, A –1 A –1, A –1, I, A, AA, AAA,…} is called the subgroup of G generated by A. (it contains A and A –1 combined with themselves any number of times.) D 4 = {I, R 90, R 180, R 270, H, V, D, D ’ } Find the subgroup of D 4 generated by each of its 8 members. THEOREM: If G is a finite group, and A is a member of G, then = { I, A, AA, AAA, AAAA, …} (this list starts repeating as soon as one of these expressions equals I, and not before). THEOREM: If G is a finite group, and A is a member of G, then = { I, A, AA, AAA, AAAA, …} (this list starts repeating as soon as one of these expressions equals I, and not before). DEFINITION: If A is a member of a finite group, then the order of A is the size of the subgroup. Find order of each member of D 4.

31. In the group C 10 = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }, find: = What is the order of each member of this group?

32. In the group C 10 = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }, find: = {0, 2, 4, 6, 8} (the evens) = {0, 3, 6, 9, 2, 5, 8, 1, 4, 7} (all of C 10 ) = {0, 4, 8, 2, 6} (the evens – the same as ) = {0, 5}. 2 has order 5 3 has order 10 4 has order 5 5 has order 2

33. In the group C 10 = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }, find: = {0, 2, 4, 6, 8} (the evens) = {0, 3, 6, 9, 2, 5, 8, 1, 4, 7} (all of C 10 ) = {0, 4, 8, 2, 6} (the evens – the same as ) = {0, 5}. 2 has order 5 3 has order 10 4 has order 5 5 has order 2

34. Product Groups

35. G 1 ×G 2 means all of the possible ways of pairing together a member of the first group with a member of the second group (wrapped in parentheses and separated by a comma) It is called the product of G 1 & G 2 EXAMPLE: If G 1 = { A, B, C } and G 2 = { 1, 2, 3, 4 }, then: G 1 ×G 2 = { (A,1), (A,2), (A,3), (A,4), (B,1), (B,2), (B,3), (B,4), (C,1), (C,2), (C,3), (C,4) }

36. G 1 ×G 2 means all of the possible ways of pairing together a member of the first group with a member of the second group (wrapped in parentheses and separated by a comma) It is called the product of G 1 & G 2 EXAMPLE: If G 1 = { A, B, C } and G 2 = { 1, 2, 3, 4 }, then: G 1 ×G 2 = { (A,1), (A,2), (A,3), (A,4), (B,1), (B,2), (B,3), (B,4), (C,1), (C,2), (C,3), (C,4) } 1234 A (A,1)(A,1)(A,2)(A,2)(A,3)(A,3)(A,4)(A,4) B (B,1)(B,1)(B,2)(B,2)(B,3)(B,3)(B,4)(B,4) C (C,1)(C,1)(C,2)(C,2)(C,3)(C,3)(C,4)(C,4) In this example G 1 ×G 2 has 12 members, because the members of can be arranged into a 3x4 table. GENERAL RULE: The size of the product of two finite groups equals the product of their sizes GENERAL RULE: The size of the product of two finite groups equals the product of their sizes

37. The Cayley table of G 1 ×G 2 Just combine the G 1 -part and the G 2 -part separately! EXAMPLE: In D 4 ×Z, compute (H,6) (V,8) = ( ??, ?? ) Recall: Z = {…, –3, –2, –1, 0, 1, 2, 3, …} = the additive group of integers.

38. The Cayley table of G 1 ×G 2 Just combine the G 1 -part and the G 2 -part separately! EXAMPLE: In D 4 ×Z, compute (H,6) (V,8) = ( R 180, 14 ) 6 + 8 = 14 in Z H*V = R 180 in D 4

39. The product of C 3 = { 0, 1, 2 } and C 2 = { 0, 1} has the following six members: C 3 ×C 2 = {(0,0), (0,1), (1,0), (1,1), (2,0), (2,1)}. What does the Cayley table look like?

40. The product of C 3 = { 0, 1, 2 } and C 2 = { 0, 1} has the following six members: C 3 ×C 2 = {(0,0), (0,1), (1,0), (1,1), (2,0), (2,1)}. Fill in the red and green separately: (0,0)(0,0)(0,1)(0,1)(1,0)(1,0)(1,1)(1,1)(2,0)(2,0)(2,1)(2,1) (0,0)(0,0)(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, ) (0,1)(0,1)(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, ) (1,0)(1,0)(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, ) (1,1)(1,1)(, )(, )(, )(, )(2,1)(2,1)(, )(, )(, )(, )(, )(, ) (2,0)(2,0)(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, ) (2,1)(2,1)(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, )(, ) (1,1) (1,0) = (2,1) because… 1+1 = 2 in C 3 and……… 1+0 = 1 in C 2.

41. The product of C 3 = { 0, 1, 2 } and C 2 = { 0, 1} has the following six members: C 3 ×C 2 = {(0,0), (0,1), (1,0), (1,1), (2,0), (2,1)}. Fill in the red and green separately: (0,0)(0,0)(0,1)(0,1)(1,0)(1,0)(1,1)(1,1)(2,0)(2,0)(2,1)(2,1) (0,0)(0,0)(0, )(0, )(0, )(0, )(1, )(1, )(1, )(1, )(2, )(2, )(2, )(2, ) (0,1)(0,1)(0, )(0, )(0, )(0, )(1, )(1, )(1, )(1, )(2, )(2, )(2, )(2, ) (1,0)(1,0)(1, )(1, )(1, )(1, )(2, )(2, )(2, )(2, )(0, )(0, )(0, )(0, ) (1,1)(1,1)(1, )(1, )(1, )(1, )(2,1)(2,1)(2, )(2, )(0, )(0, )(0, )(0, ) (2,0)(2,0)(2, )(2, )(2, )(2, )(0, )(0, )(0, )(0, )(1, )(1, )(1, )(1, ) (2,1)(2,1)(2, )(2, )(2, )(2, )(0, )(0, )(0, )(0, )(1, )(1, )(1, )(1, ) (1,1) (1,0) = (2,1) because… 1+1 = 2 in C 3 and……… 1+0 = 1 in C 2.

42. The product of C 3 = { 0, 1, 2 } and C 2 = { 0, 1} has the following six members: C 3 ×C 2 = {(0,0), (0,1), (1,0), (1,1), (2,0), (2,1)}. The Cayley table looks like this: (0,0)(0,0)(0,1)(0,1)(1,0)(1,0)(1,1)(1,1)(2,0)(2,0)(2,1)(2,1) (0,0)(0,0)(0,0)(0,0)(0,1)(0,1)(1,0)(1,0)(1,1)(1,1)(2,0)(2,0)(2,1)(2,1) (0,1)(0,1)(0,1)(0,1)(0,0)(0,0)(1,1)(1,1)(1,0)(1,0)(2,1)(2,1)(2,0)(2,0) (1,0)(1,0)(1,0)(1,0)(1,1)(1,1)(2,0)(2,0)(2,1)(2,1)(0,0)(0,0)(0,1)(0,1) (1,1)(1,1)(1,1)(1,1)(1,0)(1,0)(2,1)(2,1)(2,0)(2,0)(0,1)(0,1)(0,0)(0,0) (2,0)(2,0)(2,0)(2,0)(2,1)(2,1)(0,0)(0,0)(0,1)(0,1)(1,0)(1,0)(1,1)(1,1) (2,1)(2,1)(2,1)(2,1)(2,0)(2,0)(0,1)(0,1)(0,0)(0,0)(1,1)(1,1)(1,0)(1,0) (1,1) (1,0) = (2,1) because… 1+1 = 2 in C 3 and……… 1+0 = 1 in C 2.

43. Some symmetry groups are product groups in disguise. Explain why D 2 = the symmetry group of this cross …is isomorphic to C 2 ×C 2.

44. Some symmetry groups are product groups in disguise. Recall that C 2 = {0,1} = the symmetry group of an oriented 2-gon. +01 001 110 *NY NNY YYN Another reasonable notation is: 0 = “N” = “No rotate” 1 = “Y” = “Yes rotate” Another reasonable notation is: 0 = “N” = “No rotate” 1 = “Y” = “Yes rotate” or… Explain why D 2 = the symmetry group of this cross …is isomorphic to C 2 ×C 2. Cayley table:

45. Some symmetry groups are product groups in disguise. Explain why D 2 = the symmetry group of this cross …is isomorphic to C 2 ×C 2. I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R 180 ↔ (1,1) I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R 180 ↔ (1,1) An isomorphism: Easy to check that this dictionary translates a correct Cayley into a correct Cayley table…

46. Some symmetry groups are product groups in disguise. Explain why D 2 = the symmetry group of this cross …is isomorphic to C 2 ×C 2. I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R 180 ↔ (1,1) I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R 180 ↔ (1,1) An isomorphism: Easy to check that this dictionary translates a correct Cayley into a correct Cayley table… …but what is the visual meaning?

47. Some symmetry groups are product groups in disguise. Explain why D 2 = the symmetry group of this cross …is isomorphic to C 2 ×C 2. I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R 180 ↔ (1,1) I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R 180 ↔ (1,1) Are the ends of the green rectangle exchanged? (0=NO, 1=YES) Doesn’t exchange any ends Exchanges red ends but not green ends Exchanges green ends but not red ends Exchanges red ends and green ends An isomorphism: Are the ends of the red rectangle exchanged? (0=NO, 1=YES)

48. Some symmetry groups are product groups in disguise. Explain why D 2 = the symmetry group of this cross …is isomorphic to C 2 ×C 2. I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R 180 ↔ (1,1) I ↔ (0,0) H ↔ (1,0) V ↔ (0,1) R 180 ↔ (1,1) An isomorphism: Which viewpoint is simpler… V*H=R 180 Exchanging red ends and then green ends results in both ends being exchanged.

49. Some symmetry groups are product groups in disguise. B B B B B B B B B B B B B B B B B B B B B Explain why the symmetry group of the B-border pattern …is isomorphic to Z×C 2.

50. Some symmetry groups are product groups in disguise. Recall: Z = {…, –3, –2, –1, 0, 1, 2, 3, …} = the additive group of integers. B B B B B B B B B B B B B B B B B B B B B Explain why the symmetry group of the B-border pattern …is isomorphic to Z×C 2.

51. Some symmetry groups are product groups in disguise. Explain why the symmetry group of the B-border pattern …is isomorphic to Z×C 2. B B B B B B B B B B B B B B B B B B B B B Here are few symmetries: (T –5,YES) “translate 5 letters left and flip” (T 8,NO) “translate 8 letters right and do not flip”

52. Some symmetry groups are product groups in disguise. Explain why the symmetry group of the B-border pattern …is isomorphic to Z×C 2. B B B B B B B B B B B B B B B B B B B B B Here are few symmetries: (T –5,YES) “translate 5 letters left and flip” (T 8,NO) “translate 8 letters right and do not flip” Composing them is easy: (T –5,YES) * (T 8,NO) = (??,???).

53. Some symmetry groups are product groups in disguise. Explain why the symmetry group of the B-border pattern …is isomorphic to Z×C 2. B B B B B B B B B B B B B B B B B B B B B Here are few symmetries: (T –5,YES) “translate 5 letters left and flip” (T 8,NO) “translate 8 letters right and do not flip” Composing them is easy: (T –5,YES) * (T 8,NO) = (T 3,YES).

54. Some symmetry groups are product groups in disguise. Explain why the symmetry group of the B-border pattern …is isomorphic to Z×C 2. B B B B B B B B B B B B B B B B B B B B B Here are few symmetries: (T –5,YES) “translate 5 letters left and flip” (T 8,NO) “translate 8 letters right and do not flip” Composing them is easy: (T –5,YES) * (T 8,NO) = (T 3,YES). (–5,1) (8,0) = (3,1) We’re really working in Z×C 2 :

55. Vocabulary Review subgroup = the subgroup generated by A the order of a member of a group product group subgroup = the subgroup generated by A the order of a member of a group product group Theorem Review The size of the product of two groups equals… In a finite group, = … The size of the product of two groups equals… In a finite group, = …