Let G be a group with the property x^2 = e for all x G. Prove that G is abelian.

Question

Let G be a group with the property x^2 = e for all x G. Prove that G is abelian. Let X be a set. Prove that the power set of X, namely 2^X, is a group under the operation *, symmetric difference. Recall if A and B are sets, then A * B = (A B)(A B) = (AB) (BA). Let G be a group and fix a G. Show that the mapping lambda a : G rightarrow G, given by lambda ax = ax for x G, is a one-to-one correspondence. Show Sn is nonabelian if n > 2. Let G be a group and let ZG = {x G|xa = ax for all a G}. Prove that ZG LE G. Show that the additive group Z has infinitely many subgroups. In S3, let sigma = (12). Let H denote the subgroup generated by sigma. Give the left cosets of H. List all the cosets in Z/5Z.

Explanation / Answer

In abstract algebra, an abelian group, also called a commutativegroup, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written (the axiom of commutativity). Abelian groups generalize the arithmetic of addition of integers.

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