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

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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 l l P Zu xxy, Additive notation: H tlx l P Zu xxy. You try: What are the cyclic subgroups of S 4?

2 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 l l P Zu xxy add: H tlx l P Zu xxy.

3 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 l l P Zu xxy add: H tlx l P Zu xxy. Proposition If H xxy, then H x. Namely, (1) H n iff x n 1 and 1, x, x 2,..., x n 1 are all distinct, (2) H 8 iff x a x b for all a b.

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 l l P Zu xxy add: H tlx l P Zu xxy. Proposition If H xxy, then H x. Namely, (1) H n iff x n 1 and 1, x, x 2,..., x n 1 are all distinct, (2) H 8 iff x a x b for all a b. Theorem Any two cyclic subgroups of the same order are isomorphic. In particular, if xxy xyy ă 8, then are both cyclic groups of order n, then xxy Ñ xyy defined by x k ÞÑ y k is an isomorphism; and if xxy 8, then Z Ñ xxy defined by k ÞÑ x k is an isomorphism.

5 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 l l P Zu xxy add: H tlx l P Zu xxy. Proposition If H xxy, then H x. Namely, (1) H n iff x n 1 and 1, x, x 2,..., x n 1 are all distinct, (2) H 8 iff x a x b for all a b. Theorem Any two cyclic subgroups of the same order are isomorphic. In particular, if xxy xyy ă 8, then are both cyclic groups of order n, then xxy Ñ xyy defined by x k ÞÑ y k is an isomorphism; and if xxy 8, then Z Ñ xxy defined by k ÞÑ x k is an isomorphism. Notation: Let Z n be the cyclic group of order n (under multiplication).

6 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.

7 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.

8 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, then H xx 1 y tpx 1 q l l P Zu.

9 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, then H xx 1 y tpx 1 q l l P Zu. Proposition Let H xxy. 1. Assume x 8. Then H xx a y iff a Assume x n. Then H xx a y iff pa, nq 1. In particular, the number of generators of H is φpnq (Euler s phi function).

10 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, then H xx 1 y tpx 1 q l l P Zu. Proposition Let H xxy. 1. Assume x 8. Then H xx a y iff a Assume x n. Then H xx a y iff pa, nq 1. In particular, the number of generators of H is φpnq (Euler s phi function). Lemma 1. If x n 1 and x m 1, then x pm,nq 1.

11 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, then H xx 1 y tpx 1 q l l P Zu. Proposition Let H xxy. 1. Assume x 8. Then H xx a y iff a Assume x n. Then H xx a y iff pa, nq 1. In particular, the number of generators of H is φpnq (Euler s phi function). Lemma 1. If x n 1 and x m 1, then x pm,nq 1. Lemma 2. If x n, then x a n{pa, nq.

12 Subgroups of cyclic groups Theorem Let H xxy. 1. Every subgroup of a cyclic group is itself cyclic. (Specifically, if K ď H, then K t1u or K xx d y where d is the smallest non-negative integer such that x d P K.) 2. If H 8, then for any distinct a, b P Z ě0, 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.

13 Subgroups of cyclic groups Theorem Let H xxy. 1. Every subgroup of a cyclic group is itself cyclic. (Specifically, if K ď H, then K t1u or K xx d y where d is the smallest non-negative integer such that x d P K.) 2. If H 8, then for any distinct a, b P Z ě0, 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. This theorem says that Z 12 has exactly one subgroup of order a 1, 2, 3, 4, 6, and 12. They are 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.

14 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 ď K and there are no H ň L ň 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.

15 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 ď K and there are no H ň L ň 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.)

16 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 ď K and there are no H ň L ň 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!)

17 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 ď K and there are no H ň L ň 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.

18 Comparing lattices lattice of Z 9 xxy : xxy xx 3 y 1

19 Comparing lattices lattice of Z 9 xxy : xxy lattice of Z 3 ˆ Z 3 xxy ˆ xyy : xxy ˆ xyy xx 3 y xxy ˆ 1 xpx, yqy 1 ˆ xyy 1 1

20 Comparing lattices lattice of Z 9 xxy : xxy lattice of Z 3 ˆ Z 3 xxy ˆ xyy : xxy ˆ xyy xx 3 y xxy ˆ 1 xpx, yqy 1 ˆ xyy 1 1 Conclusion: Z 9 and Z 3 ˆ Z 3 are not isomorphic!

21 Comparing lattices lattice of Z 9 xxy : xxy lattice of Z 3 ˆ Z 3 xxy ˆ xyy : xxy ˆ xyy xx 3 y xxy ˆ 1 xpx, yqy 1 ˆ xyy 1 1 Conclusion: Z 9 and Z 3 ˆ Z 3 are not isomorphic! Thm: If G H, then G and H will have the same lattice.

22 Comparing lattices lattice of Z 9 xxy : xxy lattice of Z 3 ˆ Z 3 xxy ˆ xyy : xxy ˆ xyy xx 3 y xxy ˆ 1 xpx, yqy 1 ˆ xyy 1 1 Conclusion: Z 9 and Z 3 ˆ Z 3 are not isomorphic! Thm: If G H, then G and H will have the same lattice. The reverse is not true (consider the lattice of Z 4 ).

23 Generators and relations: two ways

24 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?

25 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 a ɛ 1 l a i P A, ɛ i 1, l P Z ě0

26 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 taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u.

27 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 taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. New definition: Consider all subgroups H ď G that contain A. Define xay is the smallest of these subgroups

28 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 taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. New definition: Consider all subgroups H ď G that contain A. Define xay is the smallest of these subgroups, i.e. A Ď xay, and if A Ď H then xay ď H.

29 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 taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. New definition: Consider all subgroups H ď G that contain A. Define xay is the smallest of these subgroups, i.e. A Ď xay, and if A Ď H then xay ď H. (Does such a thing exist??)

30 For reference: Ā ta ɛ 1 l a i P A, ɛ i 1, l P Z ě0 u.

31 For reference: Ā ta ɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. Proposition If S is a non-empty collection of subgroups of G, then the intersection of all members of S is also a subgroup of G, i.e. č H ď G. HPS

32 For reference: Ā ta ɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. Proposition If S is a non-empty collection of subgroups of G, then the intersection of all members of S is also a subgroup of G, i.e. č H ď G. HPS Definition If A is a non-empty subset of G, define the subgroup of G generated by A as xay č H. AĎH HďG

33 For reference: Ā ta ɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. Proposition If S is a non-empty collection of subgroups of G, then the intersection of all members of S is also a subgroup of G, i.e. č H ď G. HPS Definition If A is a non-empty subset of G, define the subgroup of G generated by A as xay č H. AĎH HďG Proposition In G, the set of elements generated as words in ta i, a 1 i a i P Au is the same as the smallest subgroup of G containing A, i.e. Ā xay.

34 Defining homomorphisms via generators Suppose G xay and H are groups. When does a function ϕ : A Ñ H extend to a homomorphism ϕ : G Ñ H?

35 Defining homomorphisms via generators Suppose G xay and H are groups. When does a function ϕ : A Ñ H extend to a homomorphism ϕ : G Ñ H? Namely, you know So if you know G Ā taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. 1. the images of a i for all a i P A; and 2. ϕ is a homomorphism; you know how to calculate ϕpa ɛ 1 l q ϕpa 1q ɛ 1 ϕpa 2 q ɛ2 ϕpa l q ɛ l.

36 Defining homomorphisms via generators Suppose G xay and H are groups. When does a function ϕ : A Ñ H extend to a homomorphism ϕ : G Ñ H? Namely, you know So if you know G Ā taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. 1. the images of a i for all a i P A; and 2. ϕ is a homomorphism; you know how to calculate ϕpa ɛ 1 l q ϕpa 1q ɛ 1 ϕpa 2 q ɛ2 ϕpa l q ɛ l. But is this map well-defined?

37 Defining homomorphisms via generators Suppose G xay and H are groups. When does a function ϕ : A Ñ H extend to a homomorphism ϕ : G Ñ H? Namely, you know So if you know G Ā taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. 1. the images of a i for all a i P A; and 2. ϕ is a homomorphism; you know how to calculate ϕpa ɛ 1 l q ϕpa 1q ɛ 1 ϕpa 2 q ɛ2 ϕpa l q ɛ l. But is this map well-defined? 1. ϕpgq Ď H?

38 Defining homomorphisms via generators Suppose G xay and H are groups. When does a function ϕ : A Ñ H extend to a homomorphism ϕ : G Ñ H? Namely, you know So if you know G Ā taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. 1. the images of a i for all a i P A; and 2. ϕ is a homomorphism; you know how to calculate ϕpa ɛ 1 l q ϕpa 1q ɛ 1 ϕpa 2 q ɛ2 ϕpa l q ɛ l. But is this map well-defined? 1. ϕpgq Ď H? Yes! H is a group.

39 Defining homomorphisms via generators Suppose G xay and H are groups. When does a function ϕ : A Ñ H extend to a homomorphism ϕ : G Ñ H? Namely, you know So if you know G Ā taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. 1. the images of a i for all a i P A; and 2. ϕ is a homomorphism; you know how to calculate ϕpa ɛ 1 l q ϕpa 1q ɛ 1 ϕpa 2 q ɛ2 ϕpa l q ɛ l. But is this map well-defined? 1. ϕpgq Ď H? Yes! H is a group. 2. Independent of representatives?

40 Defining homomorphisms via generators Suppose G xay and H are groups. When does a function ϕ : A Ñ H extend to a homomorphism ϕ : G Ñ H? Namely, you know So if you know G Ā taɛ 1 l a i P A, ɛ i 1, l P Z ě0 u. 1. the images of a i for all a i P A; and 2. ϕ is a homomorphism; you know how to calculate ϕpa ɛ 1 l q ϕpa 1q ɛ 1 ϕpa 2 q ɛ2 ϕpa l q ɛ l. But is this map well-defined? 1. ϕpgq Ď H? Yes! H is a group. 2. Independent of representatives? Maybe! You need to check that the relations hold!

41 Defining homomorphisms via generators Define ϕ on A and extend via ϕpa ɛ 1 l q ϕpa 1q ɛ 1 ϕpa 2 q ɛ2 ϕpa l q ɛ l. 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. ϕ : S 3 Ñ D 6 defined by p12q ÞÑ s, p23q ÞÑ sr 2. ϕ : S 3 Ñ S 5 defined by p12q ÞÑ p12q, p23q ÞÑ p23q 3. ϕ : S 3 Ñ S 2 defined by p12q ÞÑ p12q, p23q ÞÑ 1 4. ϕ : S 3 Ñ Z 2 xxy defined by p12q ÞÑ x, p23q ÞÑ x