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Suppose $X=\{x_1, x_2, \ldots, x_n\}$ is a finite set of $n$ elements. I learned that there are $n^{n^2}$ binary operations $*:X\times X \to X$ and $n^{n(n+1)/2}$ of them are commutative. I was wonder …

That $(n+1)$ divides ${2n}\choose{n}$ can be proven in different ways as done here. $$\frac{1}{n+1}\binom{2n}{n} = \binom{2n}{n} - \binom{2n}{n+1}$$ Every Catalan number $C_n=\frac{1}{n+1}\binom{2n} …

Statement: Let $G$ be a finite group. Show that if for each positive integer $m$ the number of solutions $x$ of the equation $x^n=e$ in $G$ is at most $n$, then $G$ is cyclic. There are various proo …

I was learning about Catalan numbers online. I have understood the combinatorial argument, recurrence relation and generating function based proofs of Dyck words and Dyck paths. Let $S_{2n}$ be the sy …

I read that a homomorphism is fully determined by where its generators are mapped to. Let $f\colon\, \Z\oplus \Z\to S_3$ be a group homomorphism. Generators of $\Z\oplus \Z$ are $(1,0)$ and $(0,1)$. I …