It means that the group that is presented by
$$\langle a,b\mid a^4 = 1, b^2=a^2, b^{-1}ab = a^{-1}\rangle$$
is a group with exactly $8$ elements. That is, the most general group that satisfies these relations has 8 elements.
I would suggest finding a normal form for the elements (perhaps they can all be written as $a^ib^j$ with $i$ and $j$ in certain range?) to give an upper bound, and then find a group you know that has $8$ elements, and which has two elements that satisfy the given relations and generate, to show that the group presented has at least $8$ elements (by von Dyck's theorem).
$C_{20} \times C_{30} \cong C_4 \times C_5 \times C_2 \times C_3 \times C_5$
Yes, $C_{2}$ and $C_4$ each have subgroups of order 2:
$\varphi(4) = 2$,
$\varphi(2) = 1$
"So number of elements of order 2 will be $(2 + 1)^2 - 1 = 3$, which was the correct answer."
$3$ is the correct number of subgroups of order $2$, but $3 = 2 + 1 \ne (2+1)^2 - 1 = 9 - 1 = 8$.
$\text{Aut}(C_{6125}) \cong \text{Aut}(C_{5^3}) \times \text{Aut}(C_{7^2})\not \cong C_{6125} \cong C_{5^3} \times C_{7^2}$
The automorphism group of a group is defined as a group whose elements are all the automorphisms of the base group (base group here $C_{6125}$)and where the group operation is composition of automorphisms. In other words, it gets a group structure as a subgroup of the group of all permutations of the group.
There is exactly one element of order $\,7\,$ in $\,\text{Aut}(C_{7^2})\,$ and exactly one element of order $\,5\,$ and exactly one of order $\,25\,$ in $\,\text{Aut}(C_{5^3})\,$, let's call them $\,a,\,b,\,c\,$ respectively. Then the elements of order $\,35\,$ are as follows:
$$(a^i,b^j),\;(a^i,c^{5j})\;\;\;1\leq i\leq 6,\;\;1\;\leq j\;\leq 4$$
Can you compute the number of elements of order $35$ in $\text{Aut}(C_{6125})$?
Best Answer
Definition: Let $G$ be a group and let $g\in G$. Then the order of $g$ is the smallest natural number $n$ such that $g^n = e$ (the identity element in the group). (Note that this $n$ might not exist).
So in your group, you are looking for all the elements $g$ that satisfy that
As mention in the comments below this answer, also beware that there is another notion of order in group theory. If $G$ is a finite group, then the number of elements in the group is called the order of the group. (If a group has infinitely many elements, then the group is sometimes said to have infinite order).