The question goes like this:
Write down a group table for the groups C4 and C2 x C2. For every element a in C4 and C2 x C2 determine the smallest positive integer m such that $ m\cdot a$ equals the identity element
I managed to sketch the tables for both groups and then I said something like this:
We know that the set of elements in C4 are {$ \overline 0, \overline1, \overline2 , \overline3 $} and that m $\in$ $\mathbb Z$ and m > 0.
Then for a = $\overline 1$ the right m would be 4 because m $\cdot$ a = $\overline 1$ $\cdot$ 4 = $\overline4$ = $\overline 0$ = identity element. And here m would be the smallest integer.
For a = $\overline 2$ the right m would be 2 because m $\cdot$ a = $\overline 2$ $\cdot$ 2 = $\overline4$ = $\overline 0$ = identity element. And here m would be the smallest integer.
Is this wrong? I mean to multiply an element of a group, i.e C4 with an element of a different group, i.e $\mathbb Z$? Can you guys provide me with some hints if this is wrong?