# 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.