11. Prove that a nonempty subset H of a group G is a subgroup of G...
11*. Suppose S a nonempty subset of a group G. (a) Prove that if S is finite and closed under the operation of G then S is a subgroup of G. (b) Give an example where S is closed under the group operation but S is not a subgroup.
. (15 points) Let G be a group and A be a nonempty subset of G. Consider the set Co(A) = {9 € G gag- = a for all a € A}. (a) Compute Cs, ({€, (123), (132)}), where e is the identity permutation. (b) Show that CG(A) is a subgroup of G. (c) Let H be a subgroup of G. Show that H is a subgroup of Ca(H) if and only if H is abelian.
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.
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.
#11
11. If a group G has exactly one subgroup H of order k, prove that H is normal in G. Define the centralizer of an element g in a group G to be the set 12.
Suppose H is a subset of G is a normal subgroup of index k. Prove that for any a in G, a to the power of k in H. Does this hold without the normality assumption?
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...
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....
(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.