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Chapter 2: Subgroups

2.1 Subgroups

Definition: Let GG be a group. H≤GH\leq G if HH is nonempty and HH is closed under products and inverses.
Proposition 1: (Subgroup Criterion) H≤GH\leq G if H≠∅H\neq \emptyset and xy−1∈H,∀x,y∈Hxy^{-1}\in H,\forall x,y\in H. If HH is finite, it suffices to check that H≠∅H\neq \emptyset and HH is closed under multiplication.

2.2 Centralizers, Normalizers, Stabilizers, Kernels

Definition: (Centralizer) CG(A)={g∈G∣gag−1=a,∀a∈A}C_{G}(A)=\{ g\in G \mid gag^{-1}=a, \forall a\in A \}. It is the set of elements of GG which commute with all elements of AA.
Definition: (Center) Z(G)={g∈G∣gxg−1=x,∀x∈G}Z(G)=\{ g\in G \mid gxg^{-1}=x,\forall x \in G \}. It is the set of elements of GG which commute with all elements of GG.
Definition: (Normalizer) NG(A)={g∈G∣gAg−1=A}N_{G}(A)=\{ g\in G \mid gAg^{-1}=A \}. It is the set of all elements of GG which conjugate AA.
Definition: (Stabilizer) If GG acts on SS and s∈Ss \in S is fixed, Gs={g∈G∣g⋅s=s}G_{s}=\{ g\in G \mid g\cdot s=s \}.
Definition: (Kernel) If GG acts on SS, ker={g∈G∣g⋅s=s,∀s∈S}\mathrm{ker}=\{ g\in G \mid g\cdot s=s,\forall s \in S\}.
* Note that all of these are subgroups of GG.

2.3 Cyclic Groups and Cyclic Subgroups

Definition: A group HH is cyclic if ∃x∈H\exists x \in H such that H={xn∣x∈Z}=⟨x⟩H = \{ x^{n} \mid x \in \mathbb{Z} \}=\langle x \rangle.
Proposition 2: If H=⟨x⟩H=\langle x \rangle, then ∣H∣=∣x∣\lvert H \rvert=\lvert x \rvert. Also, ∣H∣=∞  ⟹  xa≠xb\lvert H \rvert=\infty \implies x^{a}\neq x^{b}.
Proposition 3: Let GG be a group, x∈Gx \in G, and m,n∈Zm,n\in \mathbb{Z}. Then xn=xm=1  ⟹  x(m,n)=1x^{n}=x^{m}=1\implies x^{(m,n)}=1. Also, xm=1  ⟹  ∣x∣  ∣  mx^{m}=1\implies \lvert x \rvert \; \Big| \; m.
Theorem 4: Two cyclic groups of the same order are isomorphic. In particular, if ∣⟨x⟩∣=n<∞\lvert \langle x \rangle \rvert=n< \infty, ⟨x⟩≅Zn\langle x \rangle \cong Z^{n}, and if ∣⟨x⟩∣=∞\lvert \langle x \rangle \rvert=\infty, ⟨x⟩≅Z\langle x \rangle\cong \mathbb{Z}.
Proposition 5: Let GG be a group, x∈Gx \in G, and a∈Z−{0}a\in \mathbb{Z}-\{ 0 \}. x=∞  ⟹  ∣xn∣=∞∣x∣=n<∞  ⟹  ∣xa∣=n(n,a)x=\infty \implies \lvert x^{n} \rvert=\infty \qquad\lvert x \rvert=n<\infty \implies \lvert x^{a} \rvert = \frac{n}{(n,a)}.
Proposition 6: Let H=⟨x⟩H=\langle x \rangle. ∣x∣=∞  ⟹  H=⟨xa⟩  ⟺  a=±1∣x∣=n<∞  ⟹  H=⟨xa⟩  ⟺  (a,n)=1\lvert x \rvert=\infty \implies H=\langle x^{a} \rangle\iff a=\pm 1 \qquad\lvert x \rvert=n<\infty \implies H=\langle x^{a} \rangle \iff (a,n)=1.
Theorem 7: Let H=⟨x⟩H=\langle x \rangle. Then: Every subgroup of HH is cyclic. ∣H∣=∞  ⟹  ⟨xa⟩≠⟨xb⟩,∀a,b\lvert H \rvert=\infty \implies \langle x^{a} \rangle\neq \langle x^{b} \rangle,\forall a,b such that a≠ba \neq b. ∣H∣=n<∞  ⟹  ∀a\lvert H \rvert=n<\infty \implies \forall a such that a∣na \mid n there is a unique subgroup of HH of order aa that is ⟨xna⟩\left\langle x^{\frac{n}{a}} \right\rangle. Also, ∀m∈Z\forall m\in \mathbb{Z}, ⟨xm⟩=⟨x(n,m)⟩\langle x^{m} \rangle=\langle x^{(n,m)} \rangle.

2.4 Subgroups Generated by Subsets of a Group

Proposition 8: Let A\mathcal{A} be a nonempty collection of subgroups of GG. Let K=⋂H∈AHK=\bigcap_{H\in \mathcal{A}} H. Then K≤GK\leq G.
Proposition 9: A‾={a1ϵ1…anϵn∣n∈Z,n≥0,ai∈A,ϵi=±1}=⟨A⟩\overline{A}=\{ a_{1}^{\epsilon_{1}}\dots a_{n}^{\epsilon_{n}} \mid n\in \mathbb{Z}, n\geq 0,a_{i}\in A, \epsilon_{i}=\pm 1\}=\langle A \rangle.