prove: Let H be a subgroup of a group G. Then |g1H| = |g2H| for any g1, g2 ∈ G. In particular |H| = |gH| for any g ∈ G. Additionally, |Hg1| = |Hg2| and |H| = |Hg| for any g, g1, g2 ∈ G.

Abstract Algebra: 2. Let φ be an epimorphism of a finite group G1 onto a group G2 , and P a Sylow p-subgroup of G1. Prove that φ(P) is a Sylow p-subgroup of G2.

Let G be a group and D = {(x, x) | x E G}. Prove D is a subgroup of G.

|Let G and H be groups, and let 0 : G > H be a homomorphism. The set {x E G| 0(x) |e} is called the kernel of 0 and is denoted by ker (0). Show that ker (0) is a subgroup |of G -

. Let φ : G → G0 be a homomorphism of groups, and let H be a subgroup of G.

Let G1, G2 be groups such that G1 ∼= G2. Prove that if H1 ≤ G1, then there must exist some subgroup H2 of G2 such that H1 ∼= H2.

2. Let H and K be subgroups of the group G. hyk for some h e H and k e K. Show that ~ is an (a) For x, y E G, define x ~ y if x = equivalence relation on G. (b) The equivalence class of x E G is HxK coset of H and K. Show that the double cosets of H and K partition G, and that each double coset is a union of right cosets of H and is a union of left cosets of K. {hxk | h e H, k e K}. It is called a double

disprove: If G is a group and H, K < G are two subgroups, then (HK)≤ G.