Last time: Cyclic groups. Proposition

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Randolph Fowler
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1 Warmup Recall, a group H is cyclic if H can be generated by a single element. In other words, there is some element x P H for which Multiplicative notation: H tx` ` P Zu xxy, Additive notation: H t`x ` P Zu xxy. You try: What are the cyclic subgroups of S 4? Last time: Cyclic groups A group H is cyclic if H can be generated by a single element. In other words, there is some element x P H for which mult: H tx` ` P Zu xxy add: H t`x ` P Zu xxy. If H xxy, then H x. Namely, (1) H n i x n 1 and 1,x,x 2,...,x n 1 are all distinct, (2) H 8i x a x b for all a b. Theorem Any two cyclic subgroups of the same order are isomorphic. In particular, if xxy xyy 8, then are both cyclic groups of order n, then xxy Ñxyy defined by x k fiñ y k is an isomorphism; and if xxy 8, then Z Ñxxy defined by k fiñ x k is an isomorphism. Notation: Let Z n be the cyclic group of order n (under multiplication).

2 How many generators? If H xxy, (read H is generated by x, with or without relations), we call x a generator, though it generally is not the only generating element. Example: In D 8, xry xr 3 y. Example: If H xxy, thenh xx 1y tpx 1 q` ` P Zu. Let H xxy. 1. Assume x 8.ThenH xx a y i a Assume x n. ThenH xx a y i pa, nq 1. In particular, the number of generators of H is phi function). pnq (Euler s Lemma 1. If x n 1 and x m 1, thenx pm,nq 1. Lemma 2. If x n, then x a n{pa, nq.

3 Subgroups of cyclic groups Theorem Let H xxy. 1. Every subgroup of a cyclic group is itself cyclic. (Specifically, if K H, thenk t1u or K xx d y where d is the smallest non-negative integer such that x d P K.) 2. If H 8,thenforanydistincta, b P Z 0, xx a y xx b y. 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, xx m y xx pm,nq y. Example: Consider Z 12 xxy. ThistheoremsaysthatZ 12 has exactly one subgroup of order a 1, 2, 3, 4, 6, and12. Theyare a 1: 1, a 2: xx 6 y, a 3: xx 4 y xx 8 y, a 4: xx 3 y xx 9 y, a 6: xx 2 y xx 10 y, a 12 : xxy xx 5 y xx 7 y xx 11 y.

4 Subgroup lattices A subgroup lattice is a graph whose nodes are the subgroups of a group G, with an edge drawn between every pair of subgroups H, K satisfying H K and there are no H L K. Generally, we draw bigger subgroups higher than lower subgroups. Example: For G Z 12 xxy, we saw the subgroups are given by 1, xx 6 y, xx 4 y, xx 3 y, xx 2 y, and xxy G. So the subgroup lattice for Z 12 will have 6 vertices. (In general, Z n will have one subgroup/vertex for every divisor of n.) To draw the edges, you must compute the partial order of containments. (When justifying your work, show these computations!) A partial (subgroup) lattice is a graph of the relative containment relationships of a subset of the subgroups of G.

5 Comparing lattices lattice of Z 9 xxy : xxy lattice of Z 3 ˆ Z 3 xxyˆxyy : xxyˆxyy xx 3 y xxyˆ1 xpx, yqy 1 ˆxyy 1 1 Conclusion: Z 9 and Z 3 ˆ Z 3 are not isomorphic! Thm: If G H, theng and H will have the same lattice. The reverse is not true (consider the lattice of Z 4 ). Generators and relations: two ways Let G be a group, and let A be a subset of G. The relations are inherited from G, but what does A generate? We ve seen: A generates the set of elements in G... Let Ā ta 1 ` a i P A, i 1, ` P Z 0 u. New definition: Consider all subgroups H G that contain A. Define xay is the smallest of these subgroups, i.e. A Ñ xay, and if A Ñ H then xay H. (Does such a thing exist??)

6 For reference: Ā ta 1 ` a i P A, i 1, ` P Z 0 u. If S is a non-empty collection of subgroups of G, thenthe intersection of all members of S is also a subgroup of G, i.e. H G. HPS Definition If A is a non-empty subset of G, define the subgroup of G generated by A as xay AÑH H G H. In G, the set of elements generated as words in ta,a 1 i i a i P Au is the same as the smallest subgroup of G containing A, i.e. Ā xay.

7 Defining homomorphisms via generators Suppose G xay and H are groups. When does a function ' : A Ñ H extend to a homomorphism ' : G Ñ H? Namely, you know So if you know G Ā ta 1 ` a i P A, i 1, ` P Z 0 u. 1. the images of a i for all a i P A; and 2. ' is a homomorphism; you know how to calculate 'pa 1 ` q 'pa 1q 1 'pa 2 q 2 'pa`q `. But is this map well-defined? 1. 'pgq Ñ H? Yes! H is a group. 2. Independent of representatives? Maybe! You need to check that the relations hold! Defining homomorphisms via generators Define ' on A and extend via 'pa 1 ` q 'pa 1q 1 'pa 2 q 2 'pa`q `. Independent of representatives? Check relations... Example: We know S 3 xp12q, p23q p12q 2 1, p23q 2 1, p12qp23qp12q p23qp12qp23qy. Which of the following extend to homomorphisms? 1. ' : S 3 Ñ D 6 defined by p12q fiñ s, p23q fiñ sr 2. ' : S 3 Ñ S 5 defined by p12q fiñ p12q, p23q fiñ p23q 3. ' : S 3 Ñ S 2 defined by p12q fiñ p12q, p23q fiñ 1 4. ' : S 3 Ñ Z 2 xxy defined by p12q fiñ x, p23q fiñ x

Warmup Recall, a group H is cyclic if H can be generated by a single element. In other words, there is some element x P H for which Multiplicative

Warmup Recall, a group H is cyclic if H can be generated by a single element. In other words, there is some element x P H for which Multiplicative notation: H tx l l P Zu xxy, Additive notation: H tlx