(8) Let G be a group and let H be a subgroup of G. Prove that the right cosets of H partition G, that is, G= U Hy HYEH\G and, if y, y' E G and Hyn Hy' + 0, then Hy= Hy'.
7. Let G be a group and let H be a subgroup of G. Prove that the relationon G given by ab if ab-i є H is an equivalence relation.
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.
Let G be a finite group with subgroup H. Define E = { g^{-1} H g : g \in G }. Prove that |E| divides |G/H|.
Q. 5
5. Let H G be a subgroup and suppose that H,g2H.....gH are the distinet left cosets of H in G. Prove that gH - Hg for all g e G if and only if g.H Hg,, for all 2 sisr
Let G be a finite group and let H be a subgroup of G. Show using double cosets that there is a subset T of G which is simultaneously a left transversal for H and a right transversal for H.
3. a. Let H be a subgroup of a commutative group G. If every element h ∈ H is a square in H (i.e., h = k 2 for some k ∈ H), and every element of G/H is a square in G/H, then every element of G is a square in G. b. Let G be a group and H a subgroup with [G : H] = 2. If g ∈ G has odd order (i.e., ord(g) is odd),...
Only for Question3
(2) Let H be a normal subgroup of a group G. Prove that the natural operation [x][y] = [xy] gives a well-defined group structure on G/H. (3 Consider the subgroup D3 C D9. Verify that the operation from (2) is not well-defined on D9/Ds
(2) Let H be a normal subgroup of a group G. Prove that the natural operation [x][y] = [xy] gives a well-defined group structure on G/H. (3 Consider the subgroup D3 C D9....
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G
Let H be a subgroup of G. Define the normalizer of H in G to be the subset NG(H) {g € G |gHg= H}. (i) Prove that NG(H) is a subgroup of G that contains H (ii) Prove that Ha NG(H) (iii) Prove that if H < K < G, and H K, then KC NG(H)