yes, the notation "e" is fairly standard notation for the identity element (or neutral element) in an abstract group where we don't know exactly what the group operation is.
some textbooks use the notation "1" instead, but this can lead to the notion that the elements of a groups are numbers, which may not be true.
$\displaystyle e$ is the identity element.
If we assume that $\displaystyle a^m\neq e$ for $\displaystyle n\in\left\{1,\cdots,n\right\}$ then $\displaystyle a^{k_1}\neq a^{k_2}$ if $\displaystyle k_1,k_2\in \left\{1,\cdots,n\right\}$ and $\displaystyle k_1\neq k_2$. Hence $\displaystyle e,a,\cdots,a^n are$ $\displaystyle n+1$ elements of $\displaystyle e$.
consider the set of all positive powers of a: {a, a^2, a^3, a^4, a^5,....}.
can this set be infinite? if not, then wouldn't two different powers, say a^k and a^m with k ≠ m, have to be equal?
couldn't we pick k to be the bigger of the two? what would this mean a^(k-m) was equal to?