Given the following group:
$$ \left<\left\{ \begin{bmatrix}a & b \\0 & c \end{bmatrix} \mid a,b,c\in \Bbb Z_{5},a,c \neq 0 \right\} ,\:\: * \right> $$
where ∗ is multiplication.
How many sub-groups from order 2 and order 5 has here?
Which theorem I can use here to solve this?
Thanks in advance.
If $\pmatrix{a&b\cr0&c\cr}^2=I$ then $$a=4, c=4, b=0$$ $$a=4, c=1, b=0,1,2,3,4$$ $$a=1, c=4, b=0,1,2,3,4$$ therefor this group has $11$ subgroup of order $2$ and if $\pmatrix{a&b\cr0&c\cr}^5=I$ then $$a=c=1, b=1,2,3,4$$ thus this group has $4$ subgroup of order $5$.