Some properties of a finite group

Question

So I have a problem (from Cryptography course, actually) that I'm really struggling with for almost a week.

Let $G$ be a finite group, and let $e$ denote its neutral element. I need to prove the following claims:

1. There exists $E := min\{k \in \mathbb{N}: g^k = e$ for all $g \in G\}$ and $E \le |G|$.
2. If $G$ is abelian, then there exists some element in $G$ of order $E$.
3. If $K$ is a field and $G \le K^*$, then $G$ is cyclic.

I am completely new to this kind of Algebra so any hint would be highly appreciated. Also, for part 2, I have been already given a hint even though I've no clue how to use it in this context

Hint: for $a,b \in G, \langle a,b\rangle$ contains an element of order $lcm(ord(a), ord(b))$

Answer 1

I have written some questions that should help you to answer the first question (without using Langrange's theorem).

Let $g$ be an element of $G$, and consider the set $S=\{g^n\,|\,n \in \{1,...,|G|\}\}$.
(a) Noting that $S\subseteq G$, at most how many elements can $S$ have?
(b) Explain why if $S$ has the maximum number of elements, then $e$ must be in $S$.
(c) Explain why if $S$ does not have the maximum number of elements, then there exist $k, l \in \{1,...,|G|\}$ such that $1 \leq k < l \leq |G|$ and $g^l=g^k$. Show that this means that $g^{l-k} = e$ and hence $e$ is in $S$.
(d) Explain why $E \leq |G|$ (where $E$ is defined as in your post).

If this is helpful, I can do something similar for the other questions.