Abstract Algebra Answers

if a is an element of a group of finite order, prove that a^m ≠ a^n whenever m ≠ n

If M,N are R-module then M x N is also R-module

Let R be the ring and R^n= {(x1.......xn)/xi∈ R} be the R-module.

1] If I1,I2....In are ideal of R then N=I1xI2x....xIn={(x1.......xn)/xi∈ I} is a submodule in R^n

Any two disjoint permutation commute

|a|=(2*n+1);
aba^(-1)=b^(-1);
b^2=?

Use the fact that 0-a= -a to justify that
-(-5) equals 5.

*Question: Use the definition of negative numbers to justify that -(-5) equals 5.

Definition of negative numbers: -a is the number when added to a equals 0.

*Throw out and avoid the standard/traditional integer rules that you've learned and use only the knowledge we have gained working with negative numbers using the definition and number lines.
*This is a justification by proof type problem

state what properties you use/defintions/etc...

Let (G,✳) be a group, and let a∈G. Let C(a) = {g∈G: a✳g = g✳a}.

In this problem we will prove that (C(a),✳) is a subgroup of (G,✳).

C(a)⊆G by the definition of C(a).

Prove that if (G,*) is a group, and if the only subgroups of G are G and {e}, then G is cyclic.

If (H,*) is a subgroup of (G,*), and a * b is an element of H, must a is an element of H and b is an element of H? Explain your answer.