Prove that if $X,Y\triangleleft G$ and $G=XY$ that $X\cap Y=\{e\}$ then $G\cong X\times Y$.
73174
Prove $KN\cong K\times N$ if $K\cap N=\{e\}$
143979
How can I prove that $|A| + |B| = |A\cup B| + |A\cap B|$?
75604
Proof that $\partial(A\times B) = ((\partial A)\times B)\cup (A\times(\partial B))$
139833
Prove that $\gcd(M, N)\times \mbox{lcm}(M, N) = M \times N$.
61103
Prove that $\mathbb{|Q| = |Q\times Q|}$
158490
Prove that $G^{op} \cong G$ when $G^{op}$ is the opposite group of G
80892
Does $\wp(A \cap B) = \wp(A) \cap \wp(B)$ hold? How to prove it?
108213
$N_1,N_2,N_3 \unlhd G, N_i\cap N_j =\{e\}, G = N_iN_j$. Want to show that $G$ is abelian, $N_i$ are isomorphic.
115606
Proofs involving $ (A\setminus B) \cup (A \cap B)$
99346
Prove a result on transitive group actions.
129845
Prove that $T$ is a subgraph of $G$
33524
Prove the following $A \setminus (B\setminus C) = (A\setminus B)\cap (A \cup C)$
154698
Union of Cartesian products: $X \times (Y \cup Z)= (X \times Y) \cup(X\times Z)$
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
62455
Prove Equivalence Relation in G
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
118299
Show that $(\mathbb{Z}[x]/(x^{n+1}))^{\times}\cong \mathbb{Z}/2\mathbb{Z}\times\Pi_{i=1}^n\mathbb{Z}$