Is it possible to prove that if $g \in G$ with $G$ finite, then $o(g) \mid |G|$ without using Lagrange's theorem?
13807
Is there a way to prove that the order of an element in a Group divides the order of the Group, WITHOUT USING LAGRANGE'S
35124
Can every non-simple group $G$ be written as a semidirect product?
26449
$|G|=24$ prove that $G$ is not simple
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
84251
Show that $\Gamma \cup \{\neg \phi\}$ is satisfiable if and only if $\Gamma\not \models \phi$
123847
Cardinality of $GL_n(K)$ when $K$ is finite
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
145406
Prove every element of $G$ has finite order.
31252
"Efficient version" of Cayley's Theorem in Group Theory
1251
If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)
63769
Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$)
40383
Is it true that $|g| = |\phi(g)|$ for all homomorphisms $\phi: G \to G$ and $g \in G$?
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
61550
Existence of elements of order $p^k$ in a finite group
67383
Prove that $G'$ is not planar.
68422
Understanding Lagrange's Theorem (Group Theory)
8746
Groups of order $pq$ without using Sylow theorems