Pool problems in group theory
Let
Let $G$ be a group. Prove that $G$ is non-cyclic if and only if $G$ is the union of its proper subgroups.
Let
Let $G$ be a group, and $G\times G$ the direct product. The set $D=\{(g,g,)\mid g\in G\}$ is a subgroup of $G\times G$. Prove that if $D$ is normal in $G\times G$ then $G$ is abelian.
The dihedral group,
The dihedral group, $D_8$, is the group of eight rigid symmetries of a square. Prove that $D_8$ is not the internal direct product of two of its proper subgroups.
Let
Let $G$ be a finite group and $H,K\mathrel{\unlhd}G$ be normal subgroups of relatively prime order. Prove that $G$ is isomorphic to a subgroup of $G/H\times G/K$.
Suppose
Suppose $G$ is a group that contains normal subgroups $H,K\unlhd G$ with $H\cap K=\{e\}$ and $HK=G$. Prove that $G\cong H\times K$.
Let
Let $G$ be the group of upper-triangular real matrices $\begin{bmatrix} a & b \\ 0 & d\end{bmatrix}$ with $a,d\neq 0$, under matrix multiplication. Let $S$ be the subset of $G$ defined by $d=1$. Show that $S$ is normal and that $G/S\cong {\bf R}^{\times}$, the multiplicative group of nonzero real numbers.
Let
- Show that
is abelian. - Show that for any abelian group
, the inversion map is an automorphism. - Use parts (1) and (2) above to show that
is the identity element for every .
Let $G$ be a group and suppose $\operatorname{Aut}(G)$ is trivial.
\begin{enumerate}[label=(\alph*)]
\item Show that $G$ is abelian.
\item Show that for any abelian group $H$, the {\bfseries inversion map} $\phi(h)=h^{-1}$ is an automorphism.
\item Use parts (a) and (b) above to show that $g^2$ is the identity element for every $g\in G$.
\end{enumerate}
- Suppose
is a normal subgroup of a group and is the usual projection homomorphism, defined by . Prove that if is any homomorphism with , then there exists a unique homomorphism such that . (You must explicitly define , show it is well defined, show , and show that is uniquely determined.) - Prove the:
Third Isomorphism Theorem. If with , then .
\begin{enumerate}[label=\alph*)]
\item Suppose $N$ is a normal subgroup of a group $G$ and $\pi_N:G\to G/N$ is the usual projection homomorphism, defined by $\pi_N(g)=gN$. Prove that if $\phi:G\to H$ is any homomorphism with $N\leq \ker(\phi)$, then there exists a unique homomorphism $\psi:G/N\to H$ such that $\phi = \psi\circ \pi_N$. (You must explicitly define $\psi$, show it is well defined, show $\phi=\psi\circ\pi_N$, and show that $\psi$ is uniquely determined.)
\item Prove the:
\medskip
{\bfseries Third Isomorphism Theorem.} If $M, N\unlhd G$ with $N\leq M$, then $(G/N)/(M/N)\cong G/M$.
\end{enumerate}
Let
- Show that
is abelian. - Show that for any abelian group
, the inversion map is an automorphism. - Use parts (1) and (2) above to show that
is the identity element for every .
Let $G$ be a group and suppose $\operatorname{Aut}(G)$ is trivial.
\begin{enumerate}[label=(\alph*)]
\item Show that $G$ is abelian.
\item Show that for any abelian group $H$, the {\bfseries inversion map} $\phi(h)=h^{-1}$ is an automorphism.
\item Use parts (a) and (b) above to show that $g^2$ is the identity element for every $g\in G$.
\end{enumerate}
Let
contains no repetitions.
Let $G$ be a group and $a\in G$ be an element. Let $n\in {\bf N}$ be the smallest positive number such that $a^n=e$, where $e$ is the identity element. Show that the set
\[
\{e,a,a^2,\ldots, a^{n-1}\}
\]
contains no repetitions.
Let
Let $G$ be a finite abelian group of odd order. Let $\phi:G\to G$ be the function defined by $\phi(g)=g^2$ for all $g\in G$. Prove that $\phi$ is an automorphism.
Let
Let $G$ be a group with exactly two conjugacy classes. Prove that $G$ is abelian, and describe all such groups (with proof).
Let
- Prove that the map
is a well-defined automorphism of . - Prove that any automorphism of
has the form for some .
Let ${\bf Z}_n$ denote the cyclic group of order $n$. Suppose $m\in {\bf N}$ is relatively prime to $n$. Define the function $\mu_m:{\bf Z}_n\to {\bf Z}_n$ by $m[a]_n=[ma]_n$.
\begin{enumerate}[label=\alph*)]
\item Prove that the map $\mu_m$ is a well-defined automorphism of ${\bf Z}_n$.
\item Prove that any automorphism of ${\bf Z}_n$ has the form $\mu_m$ for some $m$.
\end{enumerate}
Let
You may take for granted that these are subgroups. Prove that both
Let $G$ be a finite group and $n>1$ an integer such that $(ab)^n=a^n b^n$ for all $a,b\in G$. Let
\[
G_n=\{c\in G\mid c^n=e\}\qquad\text{and}\qquad G^n=\{c^n\mid c\in G\}
\]
You may take for granted that these are subgroups. Prove that both $G_n$ and $G^n$ are normal in $G$, and $|G^n|=[G:G_n]$.
Suppose
Suppose $G$ is a group, $H\leq G$ a subgroup, and $a,b\in G$. Prove that the following are equivalent:
\begin{enumerate}[label=\alph*)]
\item $aH=bH$
\item $b\in aH$
\item $b^{-1}a\in H$
\end{enumerate}
Let
- Prove rigorously, possibly with induction, that is
, then . - Suppose
has order 5, and . Find the order of . Justify your answer.
Let $G$ be a group, and let $\operatorname{Aut}(G)$ denote the group of automorphisms of $G$. There is a homomorphism $\gamma:G\to \operatorname{Aut}(G)$ that takes $s\in G$ to the automorphism $\gamma_s$ defined by $\gamma_s(t)=sts^{-1}$.
\begin{enumerate}[label=\alph*)]
\item Prove rigorously, possibly with induction, that is $\gamma_s(t)=t^b$, then $\gamma_{s^n}(t)=t^{b^n}$.
\item Suppose $s\in G$ has order 5, and $sts^{-1}=t^2$. Find the order of $t$. Justify your answer.
\end{enumerate}
Let
- Prove that
is a subgroup of . - Prove that every non-identity element of
has infinite order. - Characterize the elements of
when , where is the additive group of real numbers.
Let $G$ be an abelian group and $G_T$ be the set of elements of finite order in $G$.
\medskip
\begin{enumerate}[label=(\alph*)]
\item Prove that $G_T$ is a subgroup of $G$.
\item Prove that every non-identity element of $G/G_T$ has infinite order.
\item Characterize the elements of $G_T$ when $G={\bf R}/{\bf Z}$, where ${\bf R}$ is the additive group of real numbers.
\end{enumerate}
Suppose
- Prove that an element in
has order dividing 2 if and only if it is its own inverse. - Prove that the number of elements in
of order 2 is odd. - Use (2) to show
must contain a subgroup of order 2.
Suppose $G$ is a finite group of even order.
\begin{enumerate}[label=\alph*)]
\item Prove that an element in $G$ has order dividing 2 if and only if it is its own inverse.
\item Prove that the number of elements in $G$ of order 2 is odd.
\item Use (b) to show $G$ must contain a subgroup of order 2.
\end{enumerate}
Let
Hint: You may use the fact that the centralizer
Let $N$ be a finite normal subgroup of $G$. Prove there is a normal subgroup $M$ of $G$ such that $[G:M]$ is finite and $nm=mn$ for all $n\in N$ and $m\in M$.
\medskip
\noindent ({\itshape Hint:} You may use the fact that the centralizer $C(h):=\{g\in G\mid ghg^{-1}=h\}$ is a subgroup of $G$.)
Suppose
Suppose $G$ is a nonempty finite set that has an associative pairing $G\times G\to G$, written $(x,y)\mapsto x\cdot y$. Suppose this pairing satisfies left and right cancellation: $x\cdot y = x\cdot y'$ implies $y=y'$, and $x\cdot y = x'\cdot y$ implies $x=x'$. Prove there exists an element $e$ of $G$ such that for all $x\in G$, $e\cdot x = x\cdot e = x$. Justify your reasoning as carefully as possible.
Show that every finite group with more than two elements has a nontrivial automorphism.
Show that every finite group with more than two elements has a nontrivial automorphism.
Let
- Prove that
is a normal subgroup of . Give an example to show that this is not true if is not onto. - Under what conditions does
induce a homomorphism , and when is this an isomorphism? Prove your answer.
Let $\sigma:G\to H$ be a group epimorphism. Let $N$ be a normal subgroup of $G$ and $K=\sigma(N)$, the image of $N$ in $H$.
\begin{enumerate}[label=\alph*)]
\item Prove that $K$ is a normal subgroup of $H$. Give an example to show that this is not true if $\sigma$ is not onto.
\item Under what conditions does $\sigma$ induce a homomorphism $G/N\to H/K$, and when is this an isomorphism? Prove your answer.
\end{enumerate}
Suppose
Suppose $G_1$ and $G_2$ are groups, with identity elements $e_1$ and $e_2$, respectively. Prove that if $\phi:G_1\to G_2$ is an isomorphism, then $\phi(e_1)=e_2$.
Suppose
- Show that the index of
is finite, and in fact . Hint: Find a set map . - Prove that equality holds in (a) if and only if
.
Suppose $A$ and $B$ are subgroups of a group $G$, and suppose $B$ is of finite index in $G$.
\begin{enumerate}[topsep=0.1in]
\item Show that the index of $A\cap B\leq A$ is finite, and in fact $|A:A\cap B|\leq |G:B|$. {\itshape Hint:} Find a set map $A/A\cap B\to G/B$.
\item Prove that equality holds in (a) if and only if $G=AB$.
\end{enumerate}
Let
Prove that
Let $G$ be a group. For each $a\in G$, let $\gamma_a$ denote the automorphism of $G$ defined by $\gamma_a(b)=aba^{-1}$ for all $b\in G$. The set $\operatorname{Inn}(G)=\{\gamma_a:a\in G\}$ is a subgroup of the automorphism group of $G$, called the subgroup of {\bfseries inner automorphisms}.
\medskip
Prove that $\operatorname{Inn}(G)$ is isomorphic to $G/Z(G)$, where $Z(G)$ is the center of $G$.
Let
Let $G$ be a group of order $2p$, where $p$ is an odd prime. Prove $G$ contains a nontrivial, proper normal subgroup.
Prove from the definition along that there are no nonabelian groups of order less than
Prove from the definition along that there are no nonabelian groups of order less than $5$.
Let
Let $G$ be a group and $H,K\mathrel{\unlhd}G$ be normal subgroups with $H\cap K=\{e\}$. Show that each element in $H$ commutes with every element in $K$.
Let
given by
- Show the operation is well defined.
- Show the operation is well defined only if the subgroup
is normal.
Let $G$ be a group and $N$ a normal subgroup of $G$. Let $aN$ denote the left coset defined by $a\in G$, and consider the binary operation
\[
G/N\times G/N\to G/N
\]
given by $(aN, bN)\mapsto abN$.
\begin{enumerate}[label=\alph*)]
\item Show the operation is well defined.
\item Show the operation is well defined only if the subgroup $N$ is normal.
\end{enumerate}
Let
Let $G$ be a group, $H\leq G$ a subgroup that is not normal. Prove there exist cosets $Ha$ and $Hb$ such that $HaHb\neq Hab$.
Let
- Prove
is a subgroup of containing . - Prove
is the largest subgroup of in which is normal.
Let $H$ be a subgroup of a group $G$. The {\bfseries normalizer} of $H$ in $G$ is the set ${\bf N}_G(H)=\{g\in G\,\mid\, gH=Hg\}$.
\begin{enumerate}[label=\alph*)]
\item Prove ${\bf N}_G(H)$ is a subgroup of $G$ containing $H$.
\item Prove ${\bf N}_G(H)$ is the largest subgroup of $G$ in which $H$ is normal.
\end{enumerate}
Let
- Prove that
is a subgroup of . - Prove that
is a normal subgroup of and that is isomorphic to a subgroup of .
Let $G$ be a group and suppose $H\leq G$. The {\bfseries normalizer} of $H$ in $G$ is defined to be $N(H)=\{g\in G\,|\, gH=Hg\}$ and the {\bfseries centralizer} of $H$ in $G$ is defined to be $C(H)=\{g\in G\,|\, gh=hg\text{ for all }h\in H\}$.
\begin{enumerate}[label=(\alph*)]
\item Prove that $N(H)$ is a subgroup of $G$.
\item Prove that $C(H)$ is a normal subgroup of $N(H)$ and that $N(H)/C(H)$ is isomorphic to a subgroup of $\operatorname{Aut}(H)$.
\end{enumerate}
Suppose
- Give a positive integer
such that . - Let
be an integer and let . Prove that the order of is .
Suppose $G$ is a cyclic group of order $n$, and $t\in G$ is a generator.
\begin{enumerate}[label=\alph*)]
\item Give a positive integer $d$ such that $t^{-1}=t^d$.
\item Let $c$ be an integer and let $m=\gcd(n,c)$. Prove that the order of $t^c$ is $\frac{n}{m}$.
\end{enumerate}
Let
Let $G$ be a finite group. Prove {\itshape from the definitions} that there exists a number $N$ such that $a^N=e$ for all $a\in G$.
Suppose
Suppose $G$ is a group and $N\unlhd G$ is a finite normal subgroup. Prove that if $G/N$ contains an element of order $n$, then $G$ also contains an element of order $n$.
Suppose
Suppose $\phi:G\to G'$ is a surjective homomorphism, $H\leq G$ is a subgroup containing $\ker(\phi)$, and $H'=\phi(H)$. Prove $\phi^{-1}(H')=H$, where $\phi^{-1}(H')=\{g\in G\,\mid\, \phi(g)\in H'\}$. Make sure to state explicitly where each hypothesis is used.
Let
Let $G$ be a group, and $H, K$ be subgroups of $G$. Let $HK=\{hk\,\mid \, h\in H, k\in K\}$ denote the set product. Prove that $HK$ is a group if and only if $HK=KH$.
Suppose
- Define a nontrivial group homomorphism
- Prove
is isomorphic to a subgroup of . - Suppose
. Prove .
Suppose $G$ is a nontrivial finite group and $H,K\mathrel{\unlhd}G$ are normal subgroups with $\gcd(|H|,|K|)=1$.
\begin{enumerate}[label=\alph*)]
\item Define a nontrivial group homomorphism $\phi:G\to G/H\times G/K$
\item Prove $G$ is isomorphic to a subgroup of $G/H\times G/K$.
\item Suppose $\gcd(m,n)=1$. Prove ${\bf Z}_{mn}\cong {\bf Z}_m\times {\bf Z}_n$.
\end{enumerate}
Suppose
- Define a group homomorphism from
to . - Compute the kernel of the homomorphism in (a), and apply the First Isomorphism Theorem.
Suppose $G$ is a group, $H$ and $K$ are normal subgroups of $G$, and $H\leq K$.
\begin{enumerate}[label=\alph*)]
\item Define a group homomorphism from $K$ to $G/H$.
\item Compute the kernel of the homomorphism in (a), and apply the First Isomorphism Theorem.
\end{enumerate}
Let
- Prove that if
is cyclic, then is abelian. - Prove that if
is nonabelian, then .
Let $G$ be a finite group and $\operatorname{Z}(G)$ denote its center.
\begin{enumerate}[label=\alph*)]
\item Prove that if $G/\operatorname{Z}(G)$ is cyclic, then $G$ is abelian.
\item Prove that if $G$ is nonabelian, then $|\operatorname{Z}(G)|\leq \frac{1}{4}|G|$.
\end{enumerate}
Let
Let $G$ be a group, $m\in {\bf N}$, and $g\in G$ be an element such that $g^m=e$. Prove that $\operatorname{o}(g)\mid m$, where $\operatorname{o}(g)$ is the order of $g$.
- Show that if
is any group (not necessarily finite) and is a subgroup, then is a disjoint union of left cosets of . - State and prove Lagrange's Theorem for finite groups.
\begin{enumerate}[label=(\alph*)]
\item Show that if $G$ is any group (not necessarily finite) and $H$ is a subgroup, then $G$ is a disjoint union of left cosets of $H$.
\item State and prove Lagrange's Theorem for finite groups.
\end{enumerate}
Let
- Prove that
is a subgroup of . - Suppose that
is normal in . Prove that .
Let $G$ be a group and $H\leq G$ a subgroup. For each coset $aH$ of $H$ in $G$, define the set
\[
G_{aH}=\{b\in G\,|\,baH=aH\}.
\]
\begin{enumerate}[label=\alph*)]
\item Prove that $G_{aH}$ is a subgroup of $G$.
\item Suppose that $H$ is normal in $G$. Prove that $G_{aH}=H$.
\end{enumerate}
Let
- Prove there exists a subgroup
of of order . - Suppose
in (a) is a normal subgroup. Prove that is contained in the center . (Recall .)
Let $G$ be a group of order $2n$ for some positive integer $n > 1$.
\begin{enumerate}[label=\alph*)]
\item Prove there exists a subgroup $K$ of $G$ of order $2$.
\item Suppose $K$ in (a) is a \underline{normal} subgroup. Prove that $K$ is contained in the center $\operatorname{Z}(G)$. (Recall $\operatorname{Z}(G)=\{a\in G\mid ab=ba\text{ for all }b\in G\}$.)
\end{enumerate}
The additive group
The additive group ${\bf Z}=({\bf Z},+)$ of rational integers is a subgroup of the additive group ${\bf Q}=({\bf Q},+)$. Show that ${\bf Z}$ has infinite index in ${\bf Q}$.
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