Theorem 2 (Recognition Theorem for Semidirect Products). Suppose G is a group and H, K
Posted: Tue Apr 26, 2022 7:48 pm
- Theorem 2 Recognition Theorem For Semidirect Products Suppose G Is A Group And H K G Are Subgroups Suppose That Hg 1 (59 KiB) Viewed 20 times
Theorem 2 (Recognition Theorem for Semidirect Products). Suppose G is a group and H, K <G are subgroups. Suppose that HG is normal, and that H NK = 1. Then HK HK, where y: K + Aut(H) corresponds to the action of K on H by conjugation (in G). In particular, if HK = G then G - H * K. = X4K
(Sylow 3) Let P E Sylp(G). (a) np =1 mod p. (6) np = [G : Ng(P)]. In particular np|m. =
Lemma 3. Let 4,8: K + Aut H be two homomorphisms, and suppose they differ by an automor- phism of K. That is, suppose there is some ye Aut(K) such that yo y = 4: K u 7 Aut H K Then H КУН x ) K. One could ask if this is the only thing that could allow different y to give different semidirect products. The answer would be no, as the following lemma shows. Lemma 4. Let 4,4 : K + Aut H be two homomorphisms, and suppose they are conjugate in Aut H. Explicitely, suppose there is some a € Aut H, corresponding to the inner automorphism Oa:B Haßa-1, and suppose that x = 0,04: Aut H K σα U Aut H Then H X4 K = H Xy K.
6. We've seen 5 groups of order 12: Z12, 26 x 22, D12, A4, and a nontrivial semidirect product 23 x 24 where the generator of Z4 acts on 23 by inverting elements. Let's prove this is all of them! (a) Let G be a group of order 12. Show that if G * A4, then G = Q * P where P is a Sylow 2-subgroup and Q is a Sylow 3-subgroup. (Hint: (Sylow 3) and the Theorem 2 should help). (b) Show that there is only one abelian and one nonabelian semidirect product Z3 x Z4 up to isomorphism. (c) Show that there is only one abelian and one nonabelian semidirect prodcut Z3 *(Z2X22) up to isomorphism. (You might need Lemma 3). (d) Put together parts (a)-(c) to deduce that there are exactly 5 groups of order 12 up to isomorphism. Of the semidirect products classified in parts (b) and (c), which one corresponds to D12?
Posted: Tue Apr 26, 2022 7:48 pm
- Theorem 2 Recognition Theorem For Semidirect Products Suppose G Is A Group And H K G Are Subgroups Suppose That Hg 1 (59 KiB) Viewed 20 times
(Sylow 3) Let P E Sylp(G). (a) np =1 mod p. (6) np = [G : Ng(P)]. In particular np|m. =
Lemma 3. Let 4,8: K + Aut H be two homomorphisms, and suppose they differ by an automor- phism of K. That is, suppose there is some ye Aut(K) such that yo y = 4: K u 7 Aut H K Then H КУН x ) K. One could ask if this is the only thing that could allow different y to give different semidirect products. The answer would be no, as the following lemma shows. Lemma 4. Let 4,4 : K + Aut H be two homomorphisms, and suppose they are conjugate in Aut H. Explicitely, suppose there is some a € Aut H, corresponding to the inner automorphism Oa:B Haßa-1, and suppose that x = 0,04: Aut H K σα U Aut H Then H X4 K = H Xy K.
6. We've seen 5 groups of order 12: Z12, 26 x 22, D12, A4, and a nontrivial semidirect product 23 x 24 where the generator of Z4 acts on 23 by inverting elements. Let's prove this is all of them! (a) Let G be a group of order 12. Show that if G * A4, then G = Q * P where P is a Sylow 2-subgroup and Q is a Sylow 3-subgroup. (Hint: (Sylow 3) and the Theorem 2 should help). (b) Show that there is only one abelian and one nonabelian semidirect product Z3 x Z4 up to isomorphism. (c) Show that there is only one abelian and one nonabelian semidirect prodcut Z3 *(Z2X22) up to isomorphism. (You might need Lemma 3). (d) Put together parts (a)-(c) to deduce that there are exactly 5 groups of order 12 up to isomorphism. Of the semidirect products classified in parts (b) and (c), which one corresponds to D12?