# Thread: Abstract Algebra: Groups, Tables, Isomorphisms

1. ## Abstract Algebra: Groups, Tables, Isomorphisms

I need help with these problems.

1. Let G be a group and f:G-->G defined by f(x)=x*x (group operation). Determine whether f is injective, surjective, bijective, or neither.

2. Let A={x element of R| x does not equal 0 or 1}. Write the table for G where G is a group of permutations of A consisting of {e(x), f(x), g(x), h(x), k(x), m(x)} defined by:
e(x)=x, f(x)=1-x, g(x)=1/x h(x)=1/(1-x) k(x)=(x-1)/x and m(x)=x/(x-1)

3. If a, b, and c are cycles of length m, n, and r, respectively, show that abc is even or odd depending on whether m + n + r - 3 is even or odd.

4. Show that the two groups G and H are isomorphic:

G=< Z2 x Z2 x Z2, +> and H={e, a, b, c, ab, ac, bc, abc} where a^2=b^2=c^2=e and (ab)^2=(bc)^2=(ac)^2=e.