11*. Suppose S a nonempty subset of a group G. (a) Prove that if S is...
11. Prove that a nonempty subset H of a group G is a subgroup of G if and only if whenever a, b E H, then ab-1 e H
. (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.
(a) Give an example of a nonempty subset U in R2 such that U is closed under multiplica- tion by scalars, but U is not a linear subspace of R2. (b) Give an example of a nonempty subset W in R2 such that W is closed under addition, but W is not a linear subspace of R2.
Real Math Analysis Let A be a nonempty finite subset of R. Prove that A is compact. Follow the comment and be serious Please. our goal is to show that we can find a finite subcover in A. However, I got stuck in finding the subcover. It is becasue finite subset means the set is bounded but it doesn't mean it is closed.
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...
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...
Let A be a nonempty subset of R. Define -A={-a: a A}. (a) Prove that if A is bounded below, then -A is bounded above. (b) Prove that if A is bounded below, then A has an infimum in R and inf A=-sup (-A).
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?
Let A be a subset of a finite group G with |A| > |G|/2. Prove that every element of G can be written as the product of two elements of A. Is this also always true when |A| = |G|/2?
1. Give an example of a group, G, and a proper subgroup, H, where H has finite index in G and H has infinite order 2. Give an example of a group, G, and a proper subgroup, H, where H has infinite index in G and H has finite order. (Hint: you won't be able to find this with the groups that we work a lot with. Try looking in SO2(R)) 1. Give an example of a group, G, and...