The terminology of the group $a^ib^j$ for $0\leq i < m$ and $0\leq j < n$, i.e. $a^m=1$ and $b^n=1$.

Question

Suppose I have a group that consists of all elements of type $a^ib^j$ for $0\leq i < m$ and $0\leq j < n$, i.e. $a^m=1$ and $b^n=1$. Here are my questions:

1) What are $a$ and $b$ called?

2) What are $m$ and $n$ called?

Any other basic related terminology I should know for this case?

Answer 1

Here $a$ and $b$ are generators of the group and $m$ and $n$ are the orders of the generators $a$ and $b$, respectively.

Assuming $a$ and $b$ commute, a potential presentation for the group is $$\langle a, b\mid a^m, b^n, ab=ba\rangle.$$