MTH3003 Final Exam Cheat Sheet

Concise formula reference for the in-class final test. Ordered by topic (matches index sections).

1. Permutations

Permutation: a Bijection $X\to X$.

• Cycle notation: $(a_1{a}_2\dots a_r)$ sends $a_i\mapsto a_{i+1}$, $a_r\mapsto a_1$, fixing the rest. Read right-to-left in products.
• Disjoint cycle decomposition: every $\sigma \in S_n$ is uniquely a product of disjoint cycles.
• Product of Permutations: feed elements through right-to-left.
• Inverse: $(a_1{a}_2\dots a_r{)}^{-1}=(a_r{a}_{r-1}\dots a_1)$.
• Order: $o(\sigma )=\mathrm{lcm}$ of the cycle lengths.

2. Groups and Subgroups

Group: closure, identity, inverses, associativity. Abelian if commutative.

subgroup: $H\subseteq G$ is a subgroup iff

1. $e\in H$, 2. $h_1{h}_2\in H$, 3. $h^{-1}\in H$.

Cyclic group: $G=⟨g⟩$ generated by one element. $|⟨g⟩|=o(g)$.

Groups generated by sets: $⟨S⟩$ = smallest subgroup containing $S$.

Klein four-group: $V_4\cong C_2\times C_2$, smallest non-cyclic group.

3. Geometric Groups

Dihedral group $D_{2n}=⟨\rho ,\sigma |{\rho }^n={\sigma }^2=e,\sigma \rho \sigma ={\rho }^{-1}⟩$ - symmetries of an $n$-gon, $|D_{2n}|=2n$.

Symmetric group $S_n$, $|S_n|=n!$. Contains every group of order $n$ (Cayley).

4. Cosets, Lagrange, Cauchy

Coset: $gH=\{gh:h\in H\}$. Cosets partition $G$, all same size $|H|$.

Lagrange’s theorem: $|H|\mid |G|$, $[G:H]=|G|/|H|$.

Cauchy’s theorem: prime $p\mid |G|\Rightarrow$ element of order $p$ exists.

Normal subgroup $H⊴G$: $g^{-1}Hg=H$ for all $g$. Detection:

• Index 2 always normal.
• Kernel of any homomorphism.
• Unique Sylow $p$-subgroup.

Quotient group $G/N$ exists for $N⊴G$, with $(g_{1}N)(g_{2}N)=(g_1{g}_2)N$. $|G/N|=|G|/|N|$.

5. Homomorphisms

Homomorphism: $\theta (gh)=\theta (g)\theta (h)$. Then $\theta (e_G)=e_H$ and $\theta (g^{-1})=\theta (g{)}^{-1}$.

Kernel $\ker (\theta )=\{g:\theta (g)=e_H\}⊴G$. Image $\mathrm{Im}(\theta )\le H$.

$\theta$ injective $⟺\ker (\theta )=\{e\}$.

Isomorphism: bijective homomorphism. $G\cong H$ means structurally identical.

Isomorphism theorems

1. $G/\ker (\theta )\cong \mathrm{Im}(\theta )$.
2. $H/(H\cap N)\cong (HN)/N$.
3. $(G/N)/(H/N)\cong G/H$ (when $N\le H$ both normal in $G$).

6. Signature & Alternating Group

Signature $\sigma (g)={\prod }_k(-1{)}^{r_k-1}$ over disjoint cycles of lengths $r_k$. Even iff number of even-length cycles is even.

Alternating group $A_n=\ker (\sigma )⊴S_n$, $|A_n|=n!/2$. Simple for $n\ge 5$.

Cycle shapes in $A_n$: $\varnothing ,(2,2),(3),(5),\dots$ (odd-length and even-pair shapes).

7. Group Actions

Group action $\lambda :G\to \mathrm{Sym}(X)$: homomorphism. Equivalently $\lambda (e)x=x$, $\lambda (gh)x=\lambda (g)(\lambda (h)x)$.

Standard actions on $X=G$:

• Regular: $\lambda (g)x=gx$. Transitive, trivial stabilisers.
• Conjugation: $\lambda (g)x=gx{g}^{-1}$. Stabiliser = centraliser. Orbits = conjugacy classes.

Orbit $Gx=\{\lambda (g)x:g\in G\}$. Stabiliser ${\mathrm{Stab}}_G(x)=\{g:\lambda (g)x=x\}\le G$. Fixed-point set ${\mathrm{Fix}}_X(g)=\{x:\lambda (g)x=x\}\subseteq X$.

Key theorems

Orbit-Stabiliser theorem: $|Gx|=|G|/|{\mathrm{Stab}}_G(x)|$.

Cayley’s theorem: every $G$ is a subgroup of $\mathrm{Sym}(G)$.

Orbit counting theorem (Burnside): $\#\text{orbits}=\frac{1}{|G|}{\sum }_g|{\mathrm{Fix}}_X(g)|$.

Colouring formula: with $c$ colours, $|{\mathrm{Fix}}_X(g)|=c^{\#\text{cycles of }g}$.

Conjugacy: $x,y$ conjugate iff $y=g^{-1}xg$ for some $g\in G$. Class = orbit under conjugation. Centraliser $C_G(x)$ = stabiliser. Class equation: $|G|=|Z(G)|+\sum |G|/|C_G(g_i)|$.

8. Sylow & Direct Products

Sylow’s theorems: write $|G|=p^{r}t$, $p\nmid t$.

1. First: Sylow $p$-subgroup of order $p^r$ exists.
2. Second: any two are conjugate; every $p$-subgroup is in some Sylow $p$-subgroup.
3. Third: $a(p)$ = number of Sylow $p$-subgroups satisfies $a(p)\mid t$ AND $a(p)\equiv 1(\mathrm{mod}p)$.

Application: $a(p)=1\Rightarrow$ unique Sylow $p$-subgroup is normal.

Internal direct product: $M,N⊴G$, $M\cap N=\{e\}$, $MN=G\Rightarrow G\cong M\times N$.

Standard pattern: orders $pq$ with $p<q$, $p\nmid q-1$ $\Rightarrow$ $G\cong C_p\times C_q\cong C_{pq}$ (cyclic). Examples: $|G|=15,33,35,51,65,77,85,221,\dots$.

Common Identities & Tricks

• $|HK|=|H||K|/|H\cap K|$ for subgroups.
• Conjugation preserves order, signature, cycle shape.
• In $S_n$: conjugacy classes = cycle shapes = integer partitions of $n$.
• Element of $G/N$ is a coset $gN$; multiplication is $(gN)(hN)=ghN$, identity is $eN=N$.
• $|G|=p^n\Rightarrow Z(G)\ne \{e\}$ (every $p$-group has non-trivial centre).
• $|G|=p^2\Rightarrow G$ abelian (either $C_{p^2}$ or $C_p\times C_p$).