2.
problem 3.
Let H be a normal subgroup of a group G and let K be any subgroup of G. Prove that the subset HK of G defined by is a subgroup of G Let G S, H ), (12) (34), (13) (24), (1 4) (23)J, and K ((13)). We know that H is a normal subgroup of S, so HK is a subgroup of S4 by Problem 2. (a) Calculate HK (b) To which familiar group is HK...
(5) Let G be a group, and let H be a subgroup of G. Define a relation ~ on G as follows: X~ · y if x-ly E H. Prove that this is an equivalence relation, and that the equivalence classes of the relation are the left cosets of H.
Let G be a finite group with subgroup H. Define E = { g^{-1} H g : g \in G }. Prove that |E| divides |G/H|.
question for 10.
(16M) Let H and K be subgroups of G. Define HK = {hk |h E H,kE K}. Suppose K is normal in G. Prove (a) HK is a subgroup of G. (b) HnK is a normal subgroup of H; K is a normal subgroup of the subgroup H K. HK K H (c) HnK
(16M) Let H and K be subgroups of G. Define HK = {hk |h E H,kE K}. Suppose K is normal in G....
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...
(K), i.e. if H is in the normalizer of K, Question 2. Let G be a group, and H,K< G. Show, if H <Norm then HK is a subgroup of G, and K is a normal subgroup of HK.
4. Let H be a subgroup of a group G and let a, b e H. Using the definition of cosets, prove that Ha= Hb if and only if ab-EH.
III. Properties of Isomorphisms. Let G and H be isomorphic groups and suppose that 0 : G + H is an isomorphism. Assume L is a subgroup of H and define K = {g € Gº(9) EL}. Prove that K is a subgroup of G.
1. Let G - Z. Let H - {0,3,5,9) be a subgroup of (you do not need to prove this is a subgroup of G). Prove that G/l is a valid quotient group. Explain what the elements of G/H are and what the group operation is. 2. Let G be a group and H a normal subgroup in G. I E H for all IEG, then prove that G/H is abelian
Let H ≤ G, and g ∈ G, Define f : H → gHg^−1 by h → ghg^−1 . Show that f is an isomorphism. Hence |gHg^−1 | = |H|.