Suppose that $G$ is a group with $n$ generators and $r$ relations whether $r < n$. Prove that $G$ is infinite.
115574
Finitely presented group with fewer relations than generators.
100605
let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$
64269
Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
133954
Does every infinite group have a maximal subgroup?
6115
Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
135701
An example of a group such that $G \cong G \times G$
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
82037
Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
134403
Let $G$ be a group, and let $n\in\mathbb Z$. Show that $(ab)^n=a^nb^n\iff(ab)^{1-n}=a^{1-n}b^{1-n}$.
148285
Generator of group $D_n$
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
14890
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
60208
Proof of property of abelian group
62455
Prove Equivalence Relation in G