Group Theory¶

Job: classify finite and finitely generated groups via actions, composition series, and abelianization; control $p$-local structure (Sylow).

Core¶

Definition. A group is a set $G$ with associative binary operation, identity $e$, and two-sided inverses. A homomorphism $\varphi:G\to H$ satisfies $\varphi(ab)=\varphi(a)\varphi(b)$; $\ker\varphi\trianglelefteq G$, $\operatorname{im}\varphi\le H$.

Notion	Definition / statement
Subgroup $H\le G$	closed under op and inverses; $e\in H$
(Left) coset	$gH$; partition of $G$; $[G:H]=\\|G/H\\|$
Lagrange	$\\|H\\|$ divides $\\|G\\|$ if $G$ finite
Normal $N\trianglelefteq G$	$gNg^{-1}=N$ $\forall g$; equiv. left=right cosets
Quotient	$G/N$ group iff $N$ normal
1st iso. thm	$G/\ker\varphi\cong\operatorname{im}\varphi$
2nd / 3rd	$HN/N\cong H/(H\cap N)$; $(G/N)/(M/N)\cong G/M$ ($N\le M\trianglelefteq G$)
Action	$\alpha:G\times X\to X$ with $\alpha(e,x)=x$, $\alpha(gh,x)=\alpha(g,\alpha(h,x))$
Orbit–stabilizer	$\\|G\cdot x\\|=[G:\operatorname{Stab}_G(x)]$
Class equation	$\|G

Exact sequences¶

A sequence $\cdots\to A\xrightarrow{f}B\xrightarrow{g}C\to\cdots$ is exact at $B$ if $\operatorname{im}f=\ker g$. Short exact $$ 0\to N\xrightarrow{\iota}G\xrightarrow{\pi}Q\to 0 $$ means $\iota$ injective, $\pi$ surjective, $\operatorname{im}\iota=\ker\pi$ (so $N\cong\ker\pi$, $Q\cong G/N$). The sequence splits if $\exists\,s:Q\to G$ with $\pi\circ s=\mathrm{id}$ (then $G\cong N\rtimes Q$ for a suitable action, or $N\times Q$ if central/trivial action as appropriate).

Sylow theory¶

Let $\|G|=p^n m$ with $p\nmid m$. A Sylow $p$-subgroup is a subgroup of order $p^n$.

Theorem (Sylow). (i) Sylow $p$-subgroups exist. (ii) Any two are conjugate. (iii) $n_p:=\#\{\text{Sylow }p\text{-subgroups}\}$ satisfies $n_p\equiv 1\pmod{p}$ and $n_p\mid m$. Moreover $n_p=[G:N_G(P)]$ for $P\in\operatorname{Syl}_p(G)$.

Structure of finitely generated abelian groups¶

Theorem. If $A$ is f.g. abelian, then $$ A\cong\mathbb{Z}^r\oplus\mathbb{Z}/d_1\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/d_k\mathbb{Z}, $$ with $d_i\ge 1$, $d_i\mid d_{i+1}$ (invariant factors), uniquely; equivalently primary decomposition $$ A\cong\mathbb{Z}^r\oplus\bigoplus_p\bigoplus_j\mathbb{Z}/p^{e_{p,j}}\mathbb{Z}. $$ Rank $r$ and the elementary divisors / invariant factors are isomorphism invariants.

Composition series¶

Theorem (Jordan–Hölder). Any two composition series of a finite group have the same length and the same multiset of composition factors up to permutation and isomorphism. Solvable $\Leftrightarrow$ composition factors cyclic of prime order.

Finite simple groups (pointer)¶

$A_n$ ($n\ge 5$), $\operatorname{PSL}_n(\mathbb{F}_q)$ (with known small exceptions), sporadic groups — classification is CFSG; for quals, know $A_5$ is simple and $A_n$ ($n\ge 5$) is simple.

See it¶

Order $pq$. Let $p<q$ primes, $p\nmid(q-1)$. Then $n_q\equiv 1\pmod q$ and $n_q\mid p$ forces $n_q=1$; similarly $n_p=1$. Thus unique (hence normal) Sylow subgroups, $G$ cyclic of order $pq$. If $p\mid(q-1)$, a non-abelian semidirect product exists (e.g. $S_3$, order $6$).

Representation theory seed. $\mathbb{C}[G]$-modules $\leftrightarrow$ complex representations; $G^{\mathrm{ab}}$ classifies $1$-dimensional characters. Maschke: char $\nmid\|G\|$ $\Rightarrow$ semisimple group algebra.

Check¶

Proof sketch: $n_p\equiv 1\pmod p$. Outline the orbit argument.

$G$ acts by conjugation on $\operatorname{Syl}_p(G)$. Stabilizer of $P$ is $N_G(P)$. Restrict the action to $P$ itself: orbits have size $1$ or divisible by $p$. Only $P$ is fixed (a $p$-group normalizing another Sylow $p$ is contained in it). Hence $n_p=1+kp$.

Counterexample: does Lagrange have a converse for composite orders?

No. $A_4$ has order $12$ but no subgroup of order $6$. (Index $2$ would be normal; analyze double cosets / element orders.)

Pitfalls¶

Cosets vs conjugates: $gHg^{-1}$ is a subgroup; $gH$ is a coset. Conjugacy of subgroups $\neq$ equality of cosets.
$H\cong K$ and $G/H\cong G/K$ do not imply $G$ determines a unique extension: group extensions are classified by $H^2(Q,N)$ (with action).
Infinite groups: Lagrange fails in the strong sense (e.g. $2\mathbb{Z}\le\mathbb{Z}$ has infinite index and infinite order); Sylow theory needs careful reformulation (pro-$p$, etc.).
Simple $\neq$ no nontrivial subgroups: simple means no nontrivial normal subgroups ($A_5$ has many subgroups).

Refs¶

Lang, Algebra (chs. on groups)
Rotman, An Introduction to the Theory of Groups
Isaacs, Finite Group Theory (character-free finite group depth)

Next¶

Rings and fields · Modules · Galois theory · Abstract algebra overview · Category theory

Notion	Definition / statement
Subgroup \(H\le G\)	closed under op and inverses; \(e\in H\)
(Left) coset	\(gH\); partition of \(G\); \([G:H]=\\|G/H\\|\)
Lagrange	\(\\|H\\|\) divides \(\\|G\\|\) if \(G\) finite
Normal \(N\trianglelefteq G\)	\(gNg^{-1}=N\) \(\forall g\); equiv. left=right cosets
Quotient	\(G/N\) group iff \(N\) normal
1st iso. thm	\(G/\ker\varphi\cong\operatorname{im}\varphi\)
2nd / 3rd	\(HN/N\cong H/(H\cap N)\); \((G/N)/(M/N)\cong G/M\) (\(N\le M\trianglelefteq G\))
Action	\(\alpha:G\times X\to X\) with \(\alpha(e,x)=x\), \(\alpha(gh,x)=\alpha(g,\alpha(h,x))\)
Orbit–stabilizer	\(\\|G\cdot x\\|=[G:\operatorname{Stab}_G(x)]\)
Class equation	$\|G