Rayat Shikshan Sanstha’s S.M.Joshi College, Hadapsar -28 Department of Mathematics Power Point Presentation Topic – Finite Groups & Subgruops Prof. Darekar S.R

Finite Groups & Subgroups

Order of a group Definition: The number of elements of a group (finite or infinite) is called its order. Notation: We will use |G| to denote the order of group G.

Examples |D4| = |Dn| = |<R90>| = |Zn| = |U(8)| = |U(11)| = |Z| = 8 2n 4 n 10 ∞

Order of an element Definition: The order of an element g in a group G is the smallest positive integer n such that gn = e (In additive notation, ng = 0). If no such integer exists, we say g has infinite order. Notation: The order of g is denoted |g|.

Examples In D4, |R90| = In D4, |H| = In Z10, |4| = In Z11, |4| = In U(8), |5| = In U(9), |5| = In Z, |1| = 4 ( R490 = R0) 2 ( H2 = R0) 5 (5•4 mod 10 = 0) 11 (11•4 mod 11 = 0) 2 (52 mod 8 = 1) 6 {5, 7, 8, 4, 2, 1} ∞ (n•1 ≠ 0 for n>0)

Group G (•mod 35) |G| = e = |5| = |10| = |15| = |20| = |30| = 6 15 1 2 3 • 5 10 15 20 25 30

Subgroups Definition: If a subset H of a group G is itself a group under the operation of G, then we say that H is a subgroup of G.

Notation We write H ≤ G to mean H is a subgroup of G. If H is not equal to G, we write H < G. We say H is a proper subgroup of G. {e} is called the trivial subgroup. All other subgroups are nontrivial.

• R0 R90 R180 R270 H V D D'

• R0 R90 R180 R270 H V D D'

• R0 R90 R180 R270 H V D D'

• R0 R90 R180 R270 H V D D'

• R0 R90 R180 R270 H V D D'

• R0 R90 R180 R270 H V D D' Note: The operation is associative, and this subset has identity and inverses The subset is not closed under *.

Subgroup tests Three important tests tell us if a nonempty subset of a group G is a subgroup of G. One-Step Subgroup Test Two-Step Subgroup Test Finite Subgroup Test

One-Step Test Let H be a nonempty subset of a group G. If ab-1 belongs to H whenever a and b belong to H, then H is a subgroup of G. (In additive groups: If a–b belongs to H whenever a and b belong to H, then H ≤ G.)

To use the One-Step Test Identify the defining property P that distinguishes elements of H. Prove the identity has property P. Assume that two elements have property P Show that ab-1 has property P. Then by the one-step test, H ≤ G. H≠ a,b in H ab-1 in H

Example - 1 Prove: Let G be an Abelian group with identity e. Let H = {x |x2 = e}. Then H ≤ G. Proof: - e2 = e, so that H is nonempty. Assume a, b in H. Then (ab-1)2 = a(b-1a)b-1 = aab-1b-1 (G is Abelian) = a2b-2 = a2(b2)-1 = ee-1 (since a and b in H) = e. By the one-step test, H ≤ G.

Example -2 Prove: The set 3Z = {3n | n in Z} (i.e. the integer multiples of 3) under the usual addition is a subgroup of Z. Proof: 0 = 3•0, so 3Z is not empty. Assume 3a and 3b are in 3Z. Then 3a – 3b = 3(a–b) is in 3Z. By the One-Step test, 3Z ≤ Z.

Two Step Test Let H be a nonempty subset of group G with operation *. If (1) H is closed under * and (2) H is closed under inverses, then H ≤ G Proof: Assume a and b are in H. By (2), b-1 is in H. By (1) ab-1 is in H. By the one-step test, H ≤ G. Sometimes it is easier to verify in two steps than in one.

Finite Subgroup Test Let H be a nonempty finite subset of a group G. If H is closed under the operation of G, then H ≤ G. Proof. Choose any a in H. By the two step test, it only remains to show that a-1 is in H.

Definition Let a be an element of a group G. The cyclic group generated by a, denoted <a> is the set of all powers of a. That is, <a> = {an | n is an integer} In additive groups, <a> = {na | n is a integer}

<a> is a subgroup Let G be group, and let a be any element of G. Then <a> is a subgroup of G. Proof: a is in <a>, so <a> is not empty. Choose any x = am and y = an in <a>. xy–1= am(an)-1 = am-n which belongs to <a> since m–n is an integer. By the one-step test, <a> is a subgroup of G.

Example - 1 • 5 10 15 20 25 30 <25> =

Example • 5 10 15 20 25 30 <25> = {25, 30, 15} Check the finite subgroup test!

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