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 C ...continues

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

cyclic groups (2)

3 more problems describing some general properties enjoyed by cyclic groups

Superincreasing Sequence; Prove that Expression is Prime

Please assist me with the attached problems including: a Superincreasing Sequence and Proving that an Expression is Prime ... Thanks!

A problem related to period(order) of an element in a group

Let a and x be elements in a group G. Prove that a and axb ,where b is the inverse of a, have the same period.

A problem related to subgroups in group theory

Let H and K be subgroups of a group G such that one is not contained in the other. Prove that H U K is not a subgroup of G.

Irreducible and one-dimensional representations

1. Classify Irreducible representations of Z over C. 2. Classify one-dimensional representations of Sn over any field k such that char k is not equal to 2.