Lecture 7 Cyclic groups and subgroups

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1 Lecture 7 Cyclic groups and subgroups

2 Review Types of groups we know Numbers: Z, Q, R, C, Q, R, C Matrices: (M n (F ), +), GL n (F ), where F = Q, R, or C. Modular groups: Z/nZ and (Z/nZ) Dihedral groups: D 2n = r, s s 2 = r n = 1, rs = sr 1 Symmetric groups: S n = { permutations of 1,..., n} Quaternians: Q 8 = i, j, k, 1 Finite groups, infinite groups, abelian groups Types of subgroups we know Kernels of homomorphisms, of group actions Images of homomorphisms The center (elements of G which commute with everything in G.) Centralizers (elements of G which commute w everything in A G) Normalizers (elements of G which setwise commute with A G) Stabilizers (given a group action on A, the elements of G which fix elements of a set S A)

3 Cyclic groups and subgroups A group H is cyclic if H can be generated by a single element. In other words, there is some element x H for which H = {x l l Z} = x.

4 Cyclic groups and subgroups A group H is cyclic if H can be generated by a single element. In other words, there is some element x H for which H = {x l l Z} = x. We call x a generator, though it generally is not the only generating element. For example, if H = x, then H = x 1 = {(x 1 ) l l Z.

5 Cyclic groups and subgroups A group H is cyclic if H can be generated by a single element. In other words, there is some element x H for which H = {x l l Z} = x. We call x a generator, though it generally is not the only generating element. For example, if H = x, then H = x 1 = {(x 1 ) l l Z. Cyclic groups can be finite or infinite: The integers are cyclic and infinite: Z = 1.

6 Cyclic groups and subgroups A group H is cyclic if H can be generated by a single element. In other words, there is some element x H for which H = {x l l Z} = x. We call x a generator, though it generally is not the only generating element. For example, if H = x, then H = x 1 = {(x 1 ) l l Z. Cyclic groups can be finite or infinite: The integers are cyclic and infinite: Z = 1. The subgroup of D 2n generated by r is cyclic and finite: The first seemingly infinite list... r n 1, r n,..., r 1, 1, r, r 2, r 3,..., r n 1, r n, r n+1,... is actually finite:... r n 1, 1,..., r n 1, 1, r, r 2, r 3,..., r n 1, 1, r,...

7 Cyclic groups and subgroups A group H is cyclic if H can be generated by a single element. In other words, there is some element x H for which H = {x l l Z} = x. We call x a generator, though it generally is not the only generating element. For example, if H = x, then H = x 1 = {(x 1 ) l l Z. Cyclic groups can be finite or infinite: The integers are cyclic and infinite: Z = 1. The subgroup of D 2n generated by r is cyclic and finite: The first seemingly infinite list... r n 1, r n,..., r 1, 1, r, r 2, r 3,..., r n 1, r n, r n+1,... is actually finite:... r n 1, 1,..., r n 1, 1, r, r 2, r 3,..., r n 1, 1, r,... In general, if a b (mod n), then r a = r b. We write H = r r n = 1.

8 Cyclic groups and subgroups A group H is cyclic if H can be generated by a single element. In other words, there is some element x H for which H = {x l l Z} = x. ( general notation) We call x a generator, though it generally is not the only generating element. For example, if H = x, then H = x 1 = {(x 1 ) l l Z. Cyclic groups can be finite or infinite: The integers are cyclic and infinite: Z = 1. The subgroup of D 2n generated by r is cyclic and finite: The first seemingly infinite list... r n 1, r n,..., r 1, 1, r, r 2, r 3,..., r n 1, r n, r n+1,... is actually finite:... r n 1, 1,..., r n 1, 1, r, r 2, r 3,..., r n 1, 1, r,... In general, if a b (mod n), then r a = r b. We write H = r r n = 1. ( specific to finite groups)

9 Proposition If H = x, then H = x.

10 Proposition If H = x, then H = x. More specifically, (1) H = n iff x n = 1 and 1, x, x 2,..., x n 1 are all distinct, (2) H = iff x a x b for all a b.

11 Proposition If H = x, then H = x. More specifically, (1) H = n iff x n = 1 and 1, x, x 2,..., x n 1 are all distinct, (2) H = iff x a x b for all a b. Proof. (Same argument as in your homework: read p. 55)

12 Proposition If H = x, then H = x. More specifically, (1) H = n iff x n = 1 and 1, x, x 2,..., x n 1 are all distinct, (2) H = iff x a x b for all a b. Proof. (Same argument as in your homework: read p. 55) Theorem Any two cyclic subgroups of the same order are isomorphic.

13 Proposition If H = x, then H = x. More specifically, (1) H = n iff x n = 1 and 1, x, x 2,..., x n 1 are all distinct, (2) H = iff x a x b for all a b. Proof. (Same argument as in your homework: read p. 55) Theorem Any two cyclic subgroups of the same order are isomorphic. In particular, 1. if n Z >0 and x and y are both cyclic groups of order n, then ϕ : x y x k y k is a well-defined bijective homomorphism,

14 Proposition If H = x, then H = x. More specifically, (1) H = n iff x n = 1 and 1, x, x 2,..., x n 1 are all distinct, (2) H = iff x a x b for all a b. Proof. (Same argument as in your homework: read p. 55) Theorem Any two cyclic subgroups of the same order are isomorphic. In particular, 1. if n Z >0 and x and y are both cyclic groups of order n, then ϕ : x y x k y k is a well-defined bijective homomorphism, or 2. if x is an infinite cyclic group, then the map ϕ : Z x k x k is a well defined bijective homomorphism. Notation: Let Z n be the cyclic group of order n.

15 How many generators? Proposition Let H = x. 1. Assume x =. Then H = x a iff a = ±1. 2. Assume x = n. Then H = x a iff (a, n) = 1. In particular, the number of generators of H is φ(n) (Euler s phi function).

16 Subgroups of cyclic groups Theorem Let H = x. 1. Every subgroup of a cyclic group is itself cyclic. (Specifically, if K H, then K = {1} or K = x d where d is the smallest non-negative integer such that x d K.) 2. If H =, then for any distinct a, b Z 0. x a x b. 3. If H = n, then for every a n, there is a unique subgroup of H of order a. ( ) This subgroup is generated by x d where d = n/a. ( ) For every m, x m = x (m,n).

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