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 \(q1\).
 Show that if \(q\) does not divide \(p1\), 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 \(p^21\) and \(p\) does not divide \(q^21\) 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 nonabelian group of order \(p^3\).
pGroups

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 \(\gcd(ba, n) = 1\).
 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 pgroups \(\implies\) 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.