Finite Groups and Subgroups, Terminology At this time we are mainly concerned with finite groups, that is, groups with a finite number of elements. The order of a group, |G|, is the number of elements in the group. The order of a group may be finite or infinite. The order of an element, |a|, is the smallest positive integer n such that an = e. The order of an element may likewise be finite or infinite. Note: if |a|=2 then a=a-1. If |a|=1 then a=e. A subgroup H of a group G is a subset of G together with the group operation, such that H is also a group. That is, H is closed under the operation, and includes inverses and identity. (Note: H must use the same group operation as G. So Zn, the integers mod n, is not a subgroup of Z, the integers, because the group operation is different.) euler portrait http://www.math.ohio-state.edu/~sinnott/ReadingClassics/homepage.html

Cancellation and Conjugation In any group, a*b=a*c implies that b=c and c*a=b*a implies that c=b. This is used in proofs. To conjugate an element a by x means to multiply thus: xax-1 or x-1ax While conjugating an element may change its value, the order |a| is preserved. This is useful in proofs and in solving matrix equations. cancellation and conjugation http://keelynet.com/indexfeb206.htm

“Socks and Shoes” Property When taking inverses of two or more elements composed together, the positions of the elements reverse. That is, (a*b)-1 = b-1*a-1. For more elements, this generalizes to (ab...yz)-1 = z-1y-1...b-1a-1. In Abelian groups, it is also true that (ab)-1 = a-1b-1 and (ab)n = anbn. This also generalizes to more elements. This is called the “socks and shoes property” as a mnemonic, because the inverse of putting on one's socks and shoes, in that order, is removing ones shoes and socks, in that order. shoes and socks in the car http://picasaweb.google.com/mp3873/PAD#5235584405081556594 shoes and socks http://www.inkfinger.us/my_weblog/2007/04/index.html

The One Step Subgroup Test Subgroup Tests: The One Step Subgroup Test Suppose G is a group and H is a non-empty subset of G. If, whenever a and b are in H, ab-1 is also in H, then H is a subgroup of G. Or, in additive notation: a - b is also in H, To apply this test: Note that H is a non-empty subset of G. Show that for any two elements a and b in H, a*b-1 is also in H. Conclude that H is a subgroup of G. ab-1 H Example: Show that the even integers are a subgroup of the Integers. Note that the even integers is not an empty set because 2 is an even integer. Let a and b be even integers. Then a = 2j and b = 2k for some integers j and k. a + (-b) = 2j + 2(-k) = 2(j-k) = an even integer Thus a - b is an even integer Thus the even integers are a subgroup of the integers. one step at a time by norby http://www.flickr.com/photos/norby/379321413/

To Apply the Two Step Subgroup Test: Subgroup Tests: The Two Step Subgroup Test Let G be a group and H a nonempty subset of G. If a●b is in H whenever a and b are in H, and a-1 is in H whenever a is in H, then H is a subgroup of G. To Apply the Two Step Subgroup Test: Note that H is nonempty Show that H is closed with respect to the group operation Show that H is closed with respect to inverses. Conclude that H is a subgroup of G. Example: show that 3Q*, the non-zero multiples of 3n where n is an integer, is a subgroup of Q*, the non-zero rational numbers. 3Q* is non-empty because 3 is an element of 3Q*. For a, b in 3Q*, a=3i and b=3j where i, j are in Q*. Then ab=3i3j=3(3ij), an element of 3Q* (closed) For a in 3Q*, a=3j for j an element in Q*. Then a-1=(j-1*3-1), an element of 3Q*. (inverses) Therefore 3Q* is a subgroup of Q*. http://www.trekearth.com/gallery/Asia/Brunei/photo653317.htm

The Finite Subgroup Test Subgroup Tests: The Finite Subgroup Test Let H be a nonempty finite subset of G. If H is closed under the group operation, then H is a subgroup of G. To Use the Finite Subgroup Test: If we know that H is finite and non-empty, all we need to do is show that H is closed under the group operation. Then we may conclude that H is a subgroup of G. Example: To Show that, in Dn, the rotations form a subgroup of Dn: Note that the set of rotations is non-empty because R0 is a rotation. Note that the composition of two rotations is always a rotation. Therefore, the rotations in Dn are a subgroup of Dn. math cartoons from http://www.math.kent.edu/~sather/ugcolloq.html

Examples of Subgroups: cyclic subgroups Let G be a group, and a an element of G. Let <a> = {an , where n is an integer}, (that is, all powers of a.) ...Or, in additive notation,,, let <a>={na, where n is an integer}, (that is, all multiples of a.) Then <a> is a subgroup of G. Note: In multiplicative notation, a0 = 1 is the identity; while 0a=0 is the identity in additive notation. Thus <a> includes the identity. Also note that the integers less than 0 are included here, so <a> includes all inverses. For example: In R*, <2>, the powers of 2, form a subgroup of R*. In Z, <2>, the even numbers, form a subgroup. In Z8, the integers mod 8, <2>={2,4,6,0} is a subgroup of Z8. In D3, the dihedral group of order 6, <R120> = {R0, R120, R240} is a subgroup of D3 Each element generates its own cyclic subgroup. subgroup image http://marauder.millersville.edu/~bikenaga/abstractalgebra/subgroup/subgroup19.png

Examples of Subgroups: The Center of a Group Z(G) The Center of a group, written Z(G), is the subset of elements in G which commute with all elements of G. If G is Abelian, then Z(G)=G. If G is non-Abelian, then Z(G) may consist only of the identity, or it may have other elements as well. For example, Z(D4) = {R0, R180}. The Center of a Group is a Subgroup of that group. Subgroup lattice for D3 http://mathworld.wolfram.com/DihedralGroupD3.html

Examples of Subgroups: The Centralizer of an Element C(a) For any element a in G, the Centralizer of a, written C(a) is the set of all elements of G which commute with a. In an Abelian group, C(a) is the entire group. In a non-Abelian group, C(a) may consist only of the identity, a, and a-1, or it may include other elements as well. For example, in D3, C(f) ={f, R0}, while C(R0)=D3 For each element a in a group G, C(a) = the centralizer of a is a subgroup of G. subgroup image http://marauder.millersville.edu/~bikenaga/abstractalgebra/subgroup/subgroup19.png