I am taking a very basic group theory course, and my teacher assigned us a homework assignment on semidihedral groups. There were four questions, and the fourth one is really stumping me.
The questions goes a follows:
Consider the Following Groups: $G$ =$\langle a, x|a^8 = x^2 = e, xax = a^3\rangle$
$H$ = $\mathbb Z_2 × \mathbb Z_8$
A homomorphism is a map, f, from a group to a group for which $f(ab) = f(a)f(b)$. Consider the map, $M$, that sends $x^b*a^c$ to the element $(b, c)$ in $H$. Is M a homomorphism? Why or why not? Write a definition of $\mathbb Z_2 × \mathbb Z_8$ in terms of generators $a$ and $b$. How do $a$ and $b$ interact in mathematical terms? i.e. what is the extra condition?
I was able to work out the first part of the question; the part about whether the map was a homomorphism or not, but the last two parts I could no figure out.
First Question: How does one use generators to define something?
Second Question: If I am able to define $\mathbb Z_2 × \mathbb Z_8$ with generators, will the answer to the second part become much clearer?
Please try to keep your answer as simple as possible so that I can actually understand what you're saying.
Thanks.
First question: The generators generate the group, in the sense that any element of the group is a product of the generators. So, if you have defined $\phi(a)$ and $\phi(b)$, then since $\phi$ is a homomorphism, there is only one choice for, say, $\phi(a^2 b^{-1} a b^2)$, since it must be $\phi(a^2 b^{-1} a b^2) = \phi(a)^2\phi(b)^{-1} \phi(a) \phi(b)^2$. As every element of the group can be written as something like this, specifying $\phi$ on the generators defines the homomorphism.
Second question: Yes.