Let Y = {gxg^-1|x is X, g in G}. Show that A=

Related BrainMass Solutions

• abelian subsets

  Therefore, C_G(S) is a subgroup in G. Second, I show that Z(G) is abelian and is a normal subgroup of G. For any x, y in Z(G), since x, y are commutative to any g in G, then x is commutative to y because y is in G. Then we have xy = yx.

• conjugate class

  So G is a p-group and must have non-trivial center C. I claim that |C| = 2. Because for any x in C, we must have gxg^(-1) = x for any g in G. Thus each element in C becomes a single conjugate class.

• Cyclic Groups

  14207 Cyclic Groups Show that every cyclic group Cn of order n is abelian. (Moreover, show that if G is a group, so is GxG) 1. Cn is a cyclic group of order n. Then Cn=<a>, where a is a generator of Cn.

• Derived subgroup proofs

  (b) Suppose x, y are in K; then g[x, y]g^{-1} = [gxg^{-1}, gyg^{-1}] belongs to K, since K is normal in G. Since K' is generated by the commutators [x, y], it follows that K' is normal in G.

• Normal subgroups

  ALSO show that <X> is normal in G if and only if gXg^-1 contained in <X> for all g belonging to G. See attached pdf file. This is a proof regarding normal subgroups.

• Isomorphism of a Group

  65231 Let G be any group, g a fixed element in G. Define phi:G--> G by phi(x) = gxg^-1. Prove that phi is an isomorphism of G onto G.

• formula

  Show that if in addition f is onto, g is one to one --- Please see the attached file. 2. Let f(x) = x2 +1 and g(x) = {x+1, x> =3; x-1, x<3 so both f and g map R into Find the formula for a.

• Normal subgroups, Second Theorem of Isomorphism, Conjugates and Cyclic Groups

  Prove that the N is a normal subgroup of G if gSg^-1<=N for all g in G d)Show that if N is cyclic, then N is normal in G if and only if for each g in G gxg^-1=x^k for some integer k. e)let n be positive integer.

• Contractible spaces

  b) Let f and g are any two maps of space X into Y. It is easy to show that f and g are homotopic.