Theorem. Let $R$ be a ring with identity, and let $a,b \in R$. Then
$$
1-ab \text{ is invertible } \iff 1-ba \text{ is invertible}.
$$

Lemma 1. Let $R$ be a ring with identity, and let $a \in R$. Suppose that $a$ has a left inverse in $R$. Then the following are equivalent:

1. $a$ has more than one left inverse.

2. $a$ is not a unit.

3. $a$ is a right zero divisor of some element.

Proposition 1. Let $a$ be an element of order $m$ in a group $G$, and let $b$ be an element of order $n$ in a group $G$. Then the element $(a,b)$ in the direct product $G \times G$ has order $[m,n]$.

Proposition 1. Any group $G$ of order $20449 = 11^2 \cdot 13^2$ must be abelian.

Proposition 1. Suppose that $G$ acts on $X$ through an action $\star$. Let $g \in G$. Define a map $f_g : X \to X$ by
$$
f_g(x) = g \star x .
$$
Then $f_g \in \operatorname{Sym}(X)$.

Definition 1. Let $\sigma\in S_n$, and write $\sigma$ as a product of disjoint cycles. We say that the form of $\sigma$ is
$$
1^{\lambda_1}2^{\lambda_2}\cdots n^{\lambda_n}
$$
if $\sigma$ has exactly $\lambda_r$ cycles of length $r$ for each $1\le r\le n$.

Example 1. The form of the permutation
$$
\sigma=(1\ 2\ 3)(4\ 5)
$$
in $S_7$ is
$$
1^22^13^14^05^06^07^0=1^22^13^1.
$$

Proposition 1. Let $N$ be a normal subgroup of a group $G$. Then $N$ is a maximal normal subgroup of $G$ if and only if $G/N$ is simple.

Lemma. Let $a$ be an element of largest order in a finite abelian group $G$, that is,
$$
o(a)=\max\{o(g)\mid g\in G\}.
$$
Then for every $g\in G$, we have
$$
o(g)\mid o(a),
$$
where $o(a)$ denotes the order of $a$. It follows that $o(a)$ is the least common multiple of the orders of all elements of $G$.

Proposition 1. If $G$ is an infinite cyclic group, then $\operatorname{Aut}(G)$ is a cyclic group of order $2$.

Proposition 1. Let $G$ be a group, and let $H$ be a normal subgroup of $G$. If the index $[G:H]=n<\infty$, then for every $x\in G$, we have $x^n\in H$.