I know how to prove a certain set is a group provided such set is basic such as $\mathbb{Z}$ under some binary operation. But I am confused with this
How to show that $\mbox{MAut}(\mathcal{C})$ and $\gamma\mbox{Aut}(\mathcal{C})$ are groups?
Where the set of monomial matrices that map a code $\mathcal{C}$ to itself forms a group of $\mbox{MAut}(\mathcal{C})$ called the monomial automorphism group of $\mathcal{C}$. The set of maps of the form $M\gamma$, where $M$ is the monomial matrix and $\gamma$ is a field automorphism, that map $\mathcal{C}$ to itself forms the group $\Gamma\mbox{Aut}(\mathcal{C})$, called automorphism group of $\mathcal{C}$.
Can someone help me with this? I would appreciate this. I don't know how the elements look like in this sets $\mbox{MAut}(\mathcal{C})$ and $\gamma\mbox{Aut}(\mathcal{C})$.