Groups.

Prove that the intersection $\bigcap_{i \in I} H_{i}$ of any family $\left\{ H_{i} \mid i \in I\right\}$ of subgroups of a group $G$ is a subgroup.

A group $G$ is said to be generated by a subset $S$ of its elements, and one writes $G=\langle S\rangle$, if the intersection of all subgroups $H$ containing $S$ coincides with $G$ (in other words, $G$ has no proper subgroup containing $S$). Show that in the case $G=\langle S\rangle$ every element $g \in G$ has the form $g=t_{1} t_{2} \ldots t_{n}, n=1,2, \ldots$, where either $t_{i} \in S$, or $t_{i}^{-1} \in S, 1 \leqslant t \leqslant n$.

Show that commuting elements $a, b$ of an arbitrary group $G$, having coprime orders $s, t$, generate in $G$ a cyclic subgroup of order $s t:\langle a, b\rangle =\langle a b\rangle$.

Show that if $M=\langle S\rangle$ is a monoid generated by a set $S$, and every element $s \in S$ is invertible in $M$, then $M$ is a group.

A group is a monoid $G$ in which equations of the form $a x=b, y a=b$ are uniquely solvable for any $a, b \in G$. Prove this statement.

Show that the set $A_{1}(\mathbb {R})$ of the so-called affine transformations $\varphi_{a, b}: x \mapsto a x+b(a, b \in \mathbb {R} ; a \neq 0)$ of the real line $\mathbb {R}$ forms a group with the multiplication law $\varphi_{a, b} \varphi_{c, d}=\varphi_{a c, a d+b}$. The group $A_{1}(\mathbb {R})$ contains the subgroup $\mathrm{GL}_{1}(\mathbb {R})$, which fixes the point $x=0$, and the subgroup of "pure translations" $x \mapsto x+b$.

The group $\mathrm{SL}_{2}(\mathbb {Z})$ contains the elements $A=\left\Vert \begin{smallmatrix} 0 & 1 \\ -1 & 0\end{smallmatrix}\right\Vert , B=\left\Vert \begin{smallmatrix} 0 & 1 \\ -1 & -1\end{smallmatrix}\right\Vert$ of orders 4 and 3 respectively. Show that $\langle A B\rangle$ is an infinite cyclic subgroup in $\mathrm{SL}_{2}(\mathbb {Z})$. Thus, the product of two elements of finite order in a group $G$ need not be an element of finite order. What is the situation in an abelian group?

Prove that a group $G$ of even order $\left|G\right|=2 n$ necessarily contains an element $g \neq e$ of order 2.

Prove that $S_{n}=\langle (12),(13), \ldots ,(1 n)\rangle$.

Prove that $S_{n}= \left\langle \left(\begin{array}{ll}1 & 2\end{array}\right),\left(\begin{array}{llll}1 & 2 & 3 & \ldots \end{array}\right)\right\rangle$.

Prove that the alternating group $A_{n}, n \geqslant 3$, is generated by cycles of length 3, and in fact

Prove that the $k$-th power $\pi^{k}$ of the cycle $\pi =\left(\begin{array}{llll}1 & 2 & \ldots & n\end{array}\right) \in S_{n}$ is a product of $d=\gcd (n, k)$ independent cycles, each of which has length $q=n / d$.

Show that the order of a permutation $\pi \in S_{n}$ (the order of the cyclic subgroup $\langle \pi \rangle$) equals the least common multiple of the lengths of the independent cycles occurring in the decomposition of $\pi$.

Let $A, B \in M_{n}(\mathbb {R})$ and $(A B)^{m}=E$ for some integer $m$. Is it true that $(B A)^{m}=E$?

Prove that a nonempty subset $H$ of a finite (multiplicative) group $G$ is a subgroup if $H$ is closed under multiplication. Hence, in this case the requirements that $H$ contain an identity element $e$ and an inverse $h^{-1}$ for each $h \in H$ are superfluous.

What system of generators can be proposed for the multiplicative group $\left(\mathbb {Q}_{+}, \cdot \right)$ of positive rational numbers?

Does there exist a finite system of generators in $\left(\mathbb {Q}_{+}, \cdot \right)$?

Prove that, up to isomorphism, there exists only a finite number $\rho (n)$ of groups of a given order $n$.

Using Exercise 10, show that every finite group can be embedded (i.e., there exists a monomorphism for it) into a finite group with two generators.

Try to verify that the diagram (Fig. 17) depicts all subgroups of the alternating group $A_{4}$. The symbol $V_{4}$ denotes the so-called

Fig. 17 four-group (or Klein group) $V_{4}=\left\{ e,(12)(34),(13)(24),(14)(23)\right\}$, and next to the other vertices of the diagram the generators of the cyclic subgroups are indicated.

Show that all groups of order 4 are abelian and, up to isomorphism, are exhausted by the permutation groups $U=\langle (1234)\rangle , V_{4}$, or else by the matrix groups:

Write out explicitly the isomorphisms $U \rightarrow L_{1}, V_{4} \rightarrow L_{2}$.