(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.
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.
(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.
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
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....
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.
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...
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.
18. Let N be a normal subgroup of a finite group G, and let Nxi, . N be a for complete list of (disjoint) right cosets. Prove that, as sets, Nx, Nz all i and j Nz,
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),...