Chapter 4

left(b a b^{-1}\right)^{n}=b a^{n} b^{-1}\(, for every positive integer \)n .\( Prove by induction. (Remember that to prove a formula such as this one by induction, you first prove it for \)n=1\(; next you prove that if it is true for \)n=k\(, then it must be true for \)n=k+1\(. You may conclude that it is true for every positive integer \)n$.)

In any finite group \(G\), the number of elements not equal to their own inverse is an even number.

\(a^{-1}\) and \(b^{-1}\) commute.

If \(a b=b a\), then \((a b)^{n}=a^{n} b^{n}\) for every positive integer \(n .\) Prove by induction.

Explain why every row of a group table must contain each element of the group exactly once. (HINT: Suppose \(x\) appears twice in the row of \(a\) : Now use the cancelation law for groups.)

If \(G\) and \(H\) are abelian, prove that \(G \times H\) is abelian.

(a b)^{2}=a^{2} b^{2}$

Suppose the groups \(G\) and \(H\) both have the following property: Every element of the group is its own inverse. Prove that \(G \times H\) also has this property.

If \(x a y=a^{-1}\), then \(y a x=a^{-1}\)

\(x a x^{-1}\) commutes with \(x b x^{-1}\), for any \(x \in G\).

(x a x)^{3}=b x \quad\( and \)\quad x^{2} a=(x a)^{-1}$

Let \(a, b\), and \(c\) each be equal to its own inverse. If \(a b=c\), then \(b c=a\) and \(c a=b .\)

Prove: \(a b=b a \quad\) iff \(\quad a b a^{-1} b^{-1}=e\)

Let \(a\) and \(b\) each be equal to its own inverse. Then \(b a\) is the inverse of \(a b\).