Counterexample for $(a*b)^2=a^2*b^2$

Question

I'm stuck on finding a counterexample for the following problem,

If $G$ is a group with operation * and $a$ and $b$ belong in $G$, then $(a*b)^2=a^2*b^2$.

I think it's false because the law of exponents only works if the group is Abelian and they never stated whether this group was Abelian. My line of thought is that I shouldn't use multiplication or addition as my operation since they are commutative but I don't know where to go from there.

Answer 1

An example from the group $(\mathbb H\setminus\{0\},\cdot)$:

$(ij)^2=k^2=-1$

but

$i^2j^2=(-1)^2=1$.

Answer 2

When constructing such examples, always try to start small. Take the smallest nonabelian group: $S_3$.

$(12), (23) \in S_3$ and $(12)(23) = (123)$.

So $[(12)(23)]^2 = (132)$. but $(12)^2 = (23)^2 = 1$, and since $(132) \not= 1$, you have the desired example.

Answer 3

Take any non-Abelian group $(G,*)$. Now take any two elements $a$ and $b$ such that $a*b\neq b*a$. Then $a*a*b*b\neq a*b*a*b$. In other words, $a^2*b^2\neq(a*b)^2$.

For instance, $G$ can be the set of all bijections from $\mathbb Q$ into itself and $*$ can be the composition. Take $a(x)=2x$ and $b(x)=1-x$.

Answer 4

$(a*b)^2 = a^2 * b^2$ if and only if $a*b*a*b=a*a*b*b$, which is if and only if $b*a=a*b$, or if and only if $a$ and $b$ commute.

Find any non-abelian group, and you will have an example.

Answer 5

Consider $D_6$, the dihedral group of order 6 the smallest non-abelian group

$ \begin{array}{c|cccccc} * & e & a & b & c & d & f \\ \hline e & e & a & b & c & d & f \\ a & a & e & d & f & b & c \\ b & b & f & e & d & c & a \\ c & c & d & f & e & a & b \\ d & d & c & a & b & f & e \\ f & f & b & c & a & e & d \\ \end{array} $

$(a*b)^2=f^2=d$

$a^2*b^2=e*e=e$

and $d\ne e$

so the property is invalid

Hope this helps

Answer 6

We know that $2 \times 2$ invertible matrices with real entries form a group under matrix multiplication.

Just try two invertible matrices and you'll probably see that their product of squares is different from the square of their product.