1. Give an example of a group, G, and a proper subgroup, H, where H has...
(10) Let G be a finite group. Prove that if H is a proper subgroup of G, then |H| = |G|/2. (11) Let G be a group. Prove that if Hį and H2 are subgroups of G such that G= H1 U H2, then either H1 = G or H2 = G.
Question 5. Give an example of a group G such that H = ·0€ G} is not a subgroup of G Remark. In class we proved that if G is an Abelian group, then H-9 shows that G being Abelian is necessary. gE G) is a subgroup of G. This question
6. Let G be a p-group and H be a proper subgroup of G. Then (1) Show that N[H]# H. (5 pts) (2) Show that N(H) is not simple. (5 pts)
Let G be a finite group, and let H be a subgroup of order n. Suppose that H is the only subgroup of order n. Show that H is normal in G. [consider the subgroup of G] aha а
Let G be a finite group with subgroup H. Define E = { g^{-1} H g : g \in G }. Prove that |E| divides |G/H|.
Exercise 2.23. Suppose H and K are subgroups of G. Prove that HK is a subgroup of G if and only if HK = KH a abaža Exercise 2.24. Suppose H is a subgroup of G. Prove that HZ(G) is a subgroup of G. Exercise 2.25. (a) Give an example of a group G with subgroups H and K such that HUK is not a subgroup of G. (b) Suppose H, H., H. ... is an infinite collection of subgroups...
4 Let G be an unknown group of order 8. By the First Sylow Theorem, G must contain a subgroup H of order 4 (a) If all subgroups of G of order 4 are isomorphic to V then what group must G be? Completely justify your answer. (b) Next, suppose that G has a subgroup H one of the following C Then G has a Cayley diagram like Find all possibilities for finishing the Cayley diagram. (c) Label each completed...
Problem 3. Subgroups of quotient groups. Let G be a group and let H<G be a normal subgroup. Let K be a subgroup of G that contains H. (1) Show that there is a well-defined injective homomorphism i: K/ H G /H given by i(kH) = kH. By abuse of notation, we regard K/H as being the subgroup Imi < G/H consisting of all cosets of the form KH with k EK. (2) Show that every subgroup of G/H is...
I help help with 34-40 33. I H is a subgroup of G and g G, prove that gHg-1 is a subgroup of G. Also, prove that the intersection of gH for all g is a normal subgroup of G. 34. Prove that 123)(min-1n-)1) 35. Prove that (12) and (123 m) generate S 36. Prove Cayley's theorem, which is the followving: Any finite group is isomorphic to a subgroup of some S 37. Let Dn be the dihedral group of...
do problems 2.25, 2.28 and 2.29 Problem 3.25. Provide an example of a group G and subgroups H and K such that HUK is not a subgroup of G Theorem 3.26. If G is a group such that H,K S G, then (HUK) S G. Moreover, (HUK) S G is the smallest subgroup containing both H and K Theorems 3.24 and 3.26 justify the use of the word "lattice" in "subgroup lattice". In general, a lattice is a partially ordered...