Semi-Direct Product and S4 Groups

Let G = (Z/3Z)^4 SemiDirectProduct S_4 be the semi-direct product of (Z/3Z)^4 and S_4. Here S_4 acts on (Z/3Z)^4 by permutating the coordinates. Hint: Given H1, H2 an element in (Z/3Z)^4 and K1, K2 an element in S4. The semi-direct product is given by the operation (H1, K1) * (H2, K2) = (H1 + K1(H2), K1 * K2) A) Find the Center of G, Z(G). B) Let phi:(Z/3Z)^4 SemiDirectProduct --> {+- 1} be the map from G to Z/2Z given by phi (h,k) = sign(k). Show this map is a homomorphism. Check file for full problem. Please provide a clear solution step by step.

Congruences

Please assist me with the attached congruence problems (hint: use Wilson's Theorem)

theory

a. Let =2 +1 (2 (Power 2(power n))) Plus 1. Prove that P is a prime Dividing , then the smallest m such that P (2 -1) is m = 2 (hint use the Division Algorithm and Binomial Theorem) Please see attached.

Theory

Suppose that... Use Lagrange's Theorem Please see attached.

cyclic groups

3 problems describing some general properties enjoyed by cyclic groups