Proof verification: For isomorphism $\phi: G\to H$, show that $|\phi (a)|=|a|$ for all $a\in G$.
29717
Not sure how to go about this group theory proof
27728
Question on a homomorphism of a set G.
11825
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
4512
Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
71819
Prove that $H$ is normal.
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
86823
proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
107511
If $G$ and $H$ are nonisomorphic group with same order then can we say that $Aut(G)$ is not isomorphic to $Aut(H)$?
138069
Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
1251
If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)
152212
Prove that the natural map $\alpha : \text{Hom}(X,\text{Spec} A) \rightarrow \text{Hom}(A,\Gamma(X,\mathcal{O}_X))$ is an isomorphism
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
84021
How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?