if $R^m \cong R^n$, then $m=n$?
43990
R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
66313
$\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$
79056
Prove that $\mathbb Z^n$ is not isomorphic to $\mathbb Z^m$ for $m\neq n$
118898
Is there a simple way of proving that $\text{GL}_n(R) \not\cong \text{GL}_m(R)$?
109532
Proving $5^n \equiv 1 \pmod {2^r}$ when $n=2^{r-2}$
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
4648
Proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$
37379
Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.
27544
With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$
61825
Congruence class $[a]$ modulo $m$, $\gcd(x, m) = \gcd(a, m)$
21673
If $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$
150958
Group theory question. If $m$ and $n$ are relatively prime, and $a$, $b$ belong to group and $a^m = b^m$ and $a^n = b^n$, how would one prove $a = b$?
78678
If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$
120495
Does this correctly interpret the "$\cong$" between groups as "$=$" between sets?
39453
How do I prove that $S^n$ is homeomorphic to $S^m \Rightarrow m=n$?
91947
Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
2043
$X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$