Abstract Algebra Answers

Let G be a group and H and K be subgroups of G of orders p and q, respectively. Prove that if p and q are relatively prime, then H intersection K = {e}.

Let G^n be the smallest subgroup of a group G that contains the set {g^n, g element of G).
Describe elements of G^n.
Prove G^n is a normal subgroup of G.
Prove that for all x element of G/G^n, x^n = e.

Let K<=H<=G (subgroups). Prove that if index (G:K) is finite, then indices (H:K) and (G:H) are also finite.

List left and right cosets of H= {epsilon, alpha} in S3, where epsilon is the identity permutation, alpha is the transposition (2,3). Is H a normal subgroup of S3? Justify.

Recursive sets are not closed under
a) complementation
b) intersection
c) substitution
d) min

Of three cards, one is painted red on both sides; one is painted black on both sides; and one is
painted red on one side and black on the other. A card is randomly chosen and placed on a table. If
the side facing up is red, what is the probability that the other-side is also red?

Prove that in any group G, orders of elements ab and ba are equal.

Let f:G1-->G2 be a homomorphism of groups. Let H be a cyclic subgroup of G1. Prove the f(H) is a cyclic subgroup of G2.

Let Ai (i ∈ I) be ideals in a ring R, and let A =(intersection on i) Ai. True or False: “If each R/Ai is von Neumann regular, then so is R/A”?

Compute the radical of the ring Tn(k) of n × n upper triangular matrices over any ring k.