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Found problems: 3

2022 District Olympiad, P1
Let $e$ be the identity of monoid $(M,\cdot)$ and $a\in M$ an invertible element. Prove that [list=a] [*]The set $M_a:=\{x\in M:ax^2a=e\}$ is nonempty; [*]If $b\in M_a$ is invertible, then $b^{-1}\in M_a$ if and only if $a^4=e$; [*]If $(M_a,\cdot)$ is a monoid, then $x^2=e$ for all $x\in M_a.$ [/list] [i]Mathematical Gazette[/i]
monoid algebra

2012 Grigore Moisil Intercounty, 4
[b]a)[/b] Prove that for any two square matrices $ A,B $ of same order the equality $ \text{ord} (AB)=\text{ord} (BA) $ is true. [b]b)[/b] Show that $ \text{ord} (ab) =\text{ord} (ba) $ if $ a,b $ are elements of a monoid and one of them is an unit.
group theory monoid algebra

2018 District Olympiad, 2
Let $p$ be a natural number greater than or equal to $2$ and let $(M, \cdot)$ be a finite monoid such that $a^p \ne a$, for any $a\in M \backslash \{e\}$, where $e$ is the identity element of $M$. Show that $(M, \cdot)$ is a group.
group monoid algebra