Suppose $G$ is a finite group, $H\leq G$, and $[G:H]=2$. Prove that for every $g\in G$, $g^2\in H$.
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
121142
Take $G$ to be a group of order $600$. Prove that for any element a in $G$ there exist a $b$ in $G$ such that $a = b^7$
111886
Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
2104
Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
5297
If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
141043
$H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
145406
Prove every element of $G$ has finite order.
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
10403
Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
101345
Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is...
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
2731
Groups of even order have an element whose order is 2.
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.
2794
Prove that if a group contains exactly one element of order 2, then that element is in the center of the group.