910 Extra Problems Group Theory

Group Theory

Centralizing and Normalizing

• Show that $C_G(H) \subseteq N_G(H) \leq G$.

• Show that $Z(G) \subseteq C_G(H) \subseteq N_G(H)$.

• Given $H\subseteq G$, let $S(H)= \bigcup_{g\in G} gHg^{-1}$, so ${\left\lvert {S(H)} \right\rvert}$ is the number of conjugates to $H$. Show that ${\left\lvert {S(H)} \right\rvert} = [G : N_G(H)]$.

  • That is, the number of subgroups conjugate to $H$ equals the index of the normalizer of $H$.

• Show that $Z(G) = \bigcap_{a\in G} C_G(a)$.

• Show that the centralizer $G_G(H)$ of a subgroup is again a subgroup.

• Show that $C_G(H) {~\trianglelefteq~}N_G(H)$ is a normal subgroup.

• Show that $C_G(G) = Z(G)$.

• Show that for $H\leq G$, $C_H(x) = H \cap C_G(x)$.

• Let $H, K \leq G$ a finite group, and without using the normalizers of $H$ or $K$, show that ${\left\lvert {HK} \right\rvert} = {\left\lvert {H} \right\rvert} {\left\lvert {K} \right\rvert}/{\left\lvert {H\cap K} \right\rvert}$.

• Show that if $H \leq N_G(K)$ then $HK \leq H$, and give a counterexample showing that this condition is necessary.

• Show that $HK$ is a subgroup of $G$ iff $HK = KH$.

• Prove that the kernel of a homomorphism is a normal subgroup.

Primes in Group Theory

• Show that any group of prime order is cyclic and simple.

• Analyze groups of order $pq$ with $q<p$.

  Hint: consider the cases when $p$ does or does not divide $q-1$.

  • Show that if $q$ does not divide $p-1$, then $G$ is cyclic.
  • Show that $G$ is never simple.

• Analyze groups of order $p^2 q$.

  Hint: Consider the cases when $q$ does or does not divide $p^2 - 1$.

• Show that no group of order $p^2 q^2$ is simple for $p<q$ primes.

• Show that a group of order $p^2 q^2$ has a normal Sylow subgroup.

• Show that a group of order $p^2 q^2$ where $q$ does not divide $p^2-1$ and $p$ does not divide $q^2-1$ is abelian.

• Show that every group of order $pqr$ with $p<q<r$ primes contains a normal Sylow subgroup.

  • Show that $G$ is never simple.

• Let $p$ be a prime and ${\left\lvert {G} \right\rvert} = p^3$. Prove that $G$ has a normal subgroup $N$ of order $p^2$.

  • Suppose $N = \left\langle{h}\right\rangle$ is cyclic and classify all possibilities for $G$ if:

    • ${\left\lvert {h} \right\rvert} = p^3$
    • ${\left\lvert {h} \right\rvert} = p$.

    Hint: Sylow and semidirect products.

• Show that any normal $p{\hbox{-}}$ subgroup is contained in every Sylow $p{\hbox{-}}$subgroup of $G$.

• Show that the order of $1+p$ in $\qty{{\mathbf{Z}}/p^2{\mathbf{Z}}}^{\times}$ is equal to $p$. Use this to construct a non-abelian group of order $p^3$.

p-Groups

• Show that every $p{\hbox{-}}$group has a nontrivial center.

• Show that every $p{\hbox{-}}$group is nilpotent.

• Show that every $p{\hbox{-}}$group is solvable.

• Show that every maximal subgroup of a $p{\hbox{-}}$group has index $p$.

• Show that every maximal subgroup of a $p{\hbox{-}}$group is normal.

• Show that every group of order $p$ is cyclic.

• Show that every group of order $p^2$ is abelian and classify them.

• Show that every normal subgroup of a $p{\hbox{-}}$group is contained in the center.

  Hint: Consider $G/Z(G)$.

• Let $O_P(G)$ be the intersection of all Sylow $p{\hbox{-}}$subgroups of $G$. Show that $O_p(G) {~\trianglelefteq~}G$, is maximal among all normal $p{\hbox{-}}$subgroups of $G$

• Let $P\in {\operatorname{Syl}}_p(H)$ where $H{~\trianglelefteq~}G$ and show that $P\cap H \in {\operatorname{Syl}}_p(H)$.

• Show that Sylow $p_i{\hbox{-}}$subgroups $S_{p_1}, S_{p_2}$ for distinct primes $p_1\neq p_2$ intersect trivially.

• Show that in a $p$ group, every normal subgroup intersects the center nontrivially.

Symmetric Groups

Specific Groups

• Show that the center of $S_3$ is trivial.
• Show that $Z(S_n) = 1$ for $n\geq 3$
• Show that $\mathop{\mathrm{Aut}}(S_3) = \mathop{\mathrm{Inn}}(S_3) \cong S_3$.
• Show that the transitive subgroups of $S_3$ are $S_3, A_3$
• Show that the transitive subgroups of $S_4$ are $S_4, A_4, D_4, {\mathbf{Z}}_2^2, {\mathbf{Z}}_4$.
• Show that $S_4$ has two normal subgroups: $A_4, {\mathbf{Z}}_2^2$.
• Show that $S_{n\geq 5}$ has one normal subgroup: $A_n$.
• $Z(A_n) = 1$ for $n\geq 4$
• Show that $[S_n, S_n] = A_n$
• Show that $[A_4, A_4] \cong {\mathbf{Z}}_2^2$
• Show that $[A_n, A_n] = A_n$ for $n\geq 5$, so $A_{n\geq 5}$ is nonabelian.

General Structure

• Show that an $m{\hbox{-}}$cycle is an odd permutation iff $m$ is an even number.
• Show that a permutation is odd iff it has an odd number of even cycles.
• Show that the center of $S_n$ for $n\geq 4$ is nontrivial.
• Show that disjoint cycles commute.
• Show directly that any $k{\hbox{-}}$cycle is a product of transpositions, and determine how many transpositions are needed.

Generating Sets

• Show that $S_n$ is generated by any of the following types of cycles:

  

  • Show that $S_n$ is generated by transpositions.
  • Show that $S_n$ is generated by adjacent transpositions.
  • Show that $S_n$ is generated by $\left\{{(12), (12\cdots n)}\right\}$ for $n\geq 2$
  • Show that $S_n$ is generated by $\left\{{(12), (23\cdots n)}\right\}$ for $n\geq 3$
  • Show that $S_n$ is generated by $\left\{{(ab), (12\cdots n)}\right\}$ where $1\leq a<b\leq n$ iff $\gcd(b-a, n) = 1$.
  • Show that $S_p$ is generated by any arbitrary transposition and any arbitrary $p{\hbox{-}}$cycle.

Alternating Groups

• Show that $A_n$ is generated $3{\hbox{-}}$cycles.
• Prove that $A_n$ is normal in $S_n$.
• Argue that $A_n$ is simple for $n \geq 5$.
• Show that $\mathop{\mathrm{Out}}(A_4)$ is nontrivial.

Dihedral Groups

• Show that if $N{~\trianglelefteq~}D_n$ is a normal subgroup of a dihedral group, then $D_n/N$ is again a dihedral group.

Other Groups

• Show that ${\mathbf{Q}}$ is not finitely generated as a group.
• Show that the Quaternion group has only one element of order 2, namely $-1$.

Classification

• Show that no group of order 36 is simple.

• Show that no group of order 90 is simple.

• Classifying all groups of order 99.

• Show that all groups of order 45 are abelian.

• Classify all groups of order 10.

• Classify the five groups of order 12.

• Classify the four groups of order 28.

• Show that if ${\left\lvert {G} \right\rvert} = 12$ and has a normal subgroup of order 4, then $G \cong A_4$.

• Suppose ${\left\lvert {G} \right\rvert} = 240 = s^4 \cdot 3 \cdot 5$.

  • How many Sylow-$p$ subgroups does $G$ have for $p\in \left\{{2, 3, 5}\right\}$?
  • Show that if $G$ has a subgroup of order 15, it has an element of order 15.
  • Show that if $G$ does not have such a subgroup, the number of Sylow-$3$ subgroups is either 10 or 40.

  Hint: Sylow on the subgroup of order 15 and semidirect products.

Group Actions

• Show that the stabilizer of an element $G_x$ is a subgroup of $G$.
• Show that if $x, y$ are in the same orbit, then their stabilizers are conjugate.
• Show that the stabilizer of an element need not be a normal subgroup?
• Show that if $G \curvearrowright X$ is a group action, then the stabilizer $G_x$ of a point is a subgroup.

Series of Groups

• Show that $A_n$ is simple for $n\geq 5$

• Give a necessary and sufficient condition for a cyclic group to be solvable.

• Prove that every simple abelian group is cyclic.

• Show that $S_n$ is generated by disjoint cycles.

• Show that $S_n$ is generated by transpositions.

• Show if $G$ is finite, then $G$ is solvable $\iff$ all of its composition factors are of prime order.

• Show that if $N$ and $G/N$ are solvable, then $G$ is solvable.

• Show that if $G$ is finite and solvable then every composition factor has prime order.

• Show that $G$ is solvable iff its derived series terminates.

• Show that $S_3$ is not nilpotent.

• Show that $G$ nilpotent $\implies G$ solvable

• Show that nilpotent groups have nontrivial centers.

• Show that Abelian $\implies$ nilpotent

• Show that p-groups $\implies$ nilpotent

Misc

• Prove Burnside’s theorem.

• Show that $\mathop{\mathrm{Inn}}(G) {~\trianglelefteq~}Aut(G)$

• Show that $\mathop{\mathrm{Inn}}(G) \cong G / Z(G)$

• Show that the kernel of the map $G\to \mathop{\mathrm{Aut}}(G)$ given by $g\mapsto (h\mapsto ghg^{-1})$ is $Z(G)$.

• Show that $N_G(H) / C_G(H) \cong A \leq Aut(H)$

• Give an example showing that normality is not transitive: i.e. $H{~\trianglelefteq~}K {~\trianglelefteq~}G$ with $H$ not normal in $G$.

Nonstandard Topics

• Show that $H~\text{char}~G \Rightarrow H \unlhd G$

  Thus “characteristic” is a strictly stronger condition than normality

• Show that $H ~\text{char}~ K ~\text{char}~G \Rightarrow H ~\text{char}~ G$

  So “characteristic” is a transitive relation for subgroups.

• Show that if $H \leq G$, $K{~\trianglelefteq~}G$ is a normal subgroup, and $H~\text{char}~K$ then $H$ is normal in $G$.

  So normality is not transitive, but strengthening one to “characteristic” gives a weak form of transitivity.