Group Theory
I began studying group theory after enrolling in Math 5105 Modern Algebra I at Missouri S&T, taught by Dr. Clayton Lungstrum. His clear and engaging teaching style made the subject both accessible and inspiring, and I consider him the best instructor I’ve had at this university. While the course provided a strong foundation, much of my deeper understanding came from self-study and working through A First Course in Abstract Algebra by John B. Fraleigh (7th Edition).
My notes include detailed proofs and explanations from the textbook, insights gained from solving problems, and explorations of challenging topics tackled independently. They also feature a dedicated section for problems with solutions and the project report where I contributed to the final proof for the Simplicity of the Alternating Group $A_n$. These sections include notes on number theory and cryptography, topics I studied independently alongside related areas in mathematics and quantum computing.
Section 1: Introductory Notes
Section 2: Fundamental Group Theory
- Definition of a Group
- Subgroup definition
- Subgroup Test
- $G\cong H\le Sym(G)$
- $D_4:$ Dihedral Group of order $8$
- $Q_8:$ Quaternion Group
- $V_4:$ Klein-$4$ Group
- $\langle X\rangle:$ Subgroup generated by a set $X$
- $H_1,H_2\le G\implies H_1\cap H_2\le G, H_1, H_2$
- Bezout's Identity
- $\langle a\rangle=\{a^n|n\in\mathbb{Z}\}$
- $o(a^s)=|\langle a^s\rangle|=\dfrac{o(a)}{gcd(s,n)}$
- $\langle a^s\rangle=\langle a^t\rangle\iff gcd(s,n)=gcd(t,n)$
- $G=\langle a^k\rangle$ when $gcd(k,n)=1$
- $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic $\iff$ $n=1,2,4,p,p^k, 2p^k$
- $(U_n,.)=\{z\in\mathbb{C}|z^n=1\}=\langle\epsilon_n\rangle\cong\mathbb{Z}/n\mathbb{Z}$
- Division Algorithm for $\mathbb{Z}$
- $H\le \langle g\rangle$, $g^n \in H (\text{least }n) \implies H = \langle g^n\rangle$
- For each divisor $d|n=|\langle a\rangle|$, $\langle a\rangle$ has exactly 1 subgroup $H_d=\langle a^{n/d}\rangle$ of order $d$
- $\mathbb{Z}_m\times \mathbb{Z}_n\cong \mathbb{Z}_{mn}$ and cyclic$ \iff gcd(m,n)=1$
- $G=\langle x_1,x_2,\cdots,x_s\rangle$ is abelian $\implies$ $G\cong \mathbb{Z}^r\times\mathbb{Z}_{p_1^{\alpha_1}}\times\mathbb{Z}_{p_2^{\alpha_2}}\times\cdots\times\mathbb{Z}_{p_n^{\alpha_n}}$
- Direct Product
- Internal Direct Product
- Cosets
- $xH=yH$ or $xH\cap yH=\phi$
- $G=\cup_{g\in G}gH$
- $|gH|=|H|$
- Lagrange's Theorem
- $K\le H\le G\implies [G:K]=[G:H][H:K]$
- If $G$ is finite and $x\in G$, then $o(x)| |G|$
Section 3: Factor Groups and Homomorphisms
- Homomorphisms
- $1'=\phi(1), \phi(a^{-1})=\phi(a)^{-1}, \phi(H)\le G'$
- Isomorphism Test
- Inn($G$) $\trianglelefteq$ Aut($G$) $\le$ Sym($G$)
- $gHg^{-1}:$ Conjugate Subgroup of $H$
- $H$ char $G\iff \sigma(H)=H\forall\sigma\in$Aut($G$)$\implies H\trianglelefteq G$
- $C_G(g)\le G:$ Centralizer of $g$ in $G$
- $Z(G)\trianglelefteq G$, $G/Z(G)$ cyclic $\implies$ $G$ is abelian $\big(Z(G)=G\big)$
- $G'=\langle[x,y]=xyx^{-1}y^{-1}|x,y\in G\rangle:$ Commutator Subgroup of G
- core$_G(H)=\cap_{g\in G}gHg^{-1}\trianglelefteq G:$ largest normal subgroup of $G$ that is contained in $G$
- Normal Subgroup of $G$
- $N\trianglelefteq G\implies \phi(N)\trianglelefteq\phi(G)$
- $G$ has exactly 1 subgroup $H$ of a given order $\implies H\trianglelefteq G$
- $H\le G\&[G:H]=2\implies H\trianglelefteq G$
- $\langle HN\rangle:$ Join of $H$ and $N$
- $HK\le G\iff HK=KH$
- $H\le G,$ $N\trianglelefteq G\implies HN\le G$ | $|HK|=\dfrac{|H|.|K|}{|H\cap K|}$
- $ker(\phi)\trianglelefteq G$
- $G/H:$ Factor Group
- $|G/H|=[G:H]=\dfrac{|G|}{|H|}$
- $\gamma:G\to G/H,\gamma(x)=xH$ is a group homomorphism with $ker(\gamma)=H$
- First Isomorphism Theorem
- $K\le G/N\implies K=H/N$ for some $H$ st $N\le H\le G$
- $G=\langle a\rangle\implies G/H=\langle aH\rangle$
- Second Isomorphism Theorem
- Third Isomorphism Theorem
Section 4: Series of Groups
- Normal(Subnormal) Series
- Refinement of a Series
- Butterfly Lemma
- Schreier Theorem
- Simple Groups
- $G$ has exactly $2$ subgroups$\implies G\cong \mathbb{Z}_p$ is cyclic of prime order
- $M$ is maximal subgroup of $G$ $\&$ $M\trianglelefteq G\implies |G/M|$ is prime
- $G$ simple, abelian $\iff$ $G$ simple, cyclic $\iff$ $|G|$ prime
- $M$ is maximal normal subgroup of $G$ $\iff$ $G/M$ is simple
- Simple Groups as Building Blocks of Finite Groups
- Composition and Principal Series
- Jordan-Holder Theorem
- Solvable Groups
- Nilpotent Group
- A Finite $p$-group is Nilpotent
Section 5: Group Action on a Set
- G-Sets, Faithful Action
- Regular Action, Conjugation Action
- n!-theorem
- $[G:H]=p,$ least prime divisor of $|G|\implies H\trianglelefteq G$
- $[G:H]=2\implies H\trianglelefteq G$
- $G_x\le G:$ Stabilizer of $x$
- $C_G(x)/N_G(H):$ Stabilizer under conjugation of $x/H$
- $Gx:$ Orbit of $x$
- FCP: $|Gx|=[G:G_x]$
- Landau's Theorem
- $HgK:$ Double Coset of $g$
- $|HK|=\frac{|H|.|K|}{|H\cap K|}$
- $|G|=p^n\implies Z(G)>1$
- $|G|=p^2\implies$ $G$ is abelian $\big( Z(G)=G \big)$
- Burnside/Cauchy-Frobenius Lemma
Section 6: Sylow Theorems
- Introduction
- $|X|=\sum_{i=1}^{r}|Gx_i|=|X_G|+\sum_{i=s+1}^r|Gx_i|$
- $p$-group definition
- Cauchy's Theorem
- $G$ finite, then $G$ is $p$-group $\iff$ $|G|=p^n$
- $P$ is $p$-subgroup, $Q$ is $q$-subgroup $\implies$ $P\cap Q=1$
- Sylow $p-$subgroup of $G$
- Normalizer of a Subgroup
- First Sylow Theorem
- Second Sylow Theorem
- $n_p=|Syl_p(G)|=[G:N_G(P)]$
- Third Sylow Theorem
- $|G|=pq,p>q\implies$ $G$ has a normal Sylow $p$-subgroup (n_p=1)
- $|G|=p^2q\implies n_p=1$ or $n_q=1\implies G$ is not simple
- Burnside's Theorem
- Frattini Argument
- $\Phi(G)=\cap_{M\text{ maximal in }G}M\trianglelefteq G:$ Frattini Subgroup of $G$
Section 7: Symmetric Groups
- Elements of $S_n$ are the permutations
- Orbit of an element under $\sigma\in S_n$
- $k$-cycle
- $\sigma=$ product of disjoint cycles (unique up to the order of cycles)
- Cycle Structure of a Permutation
- $\sigma,\tau\in S_n$ are conjugate iff same cycle structure
- $\sigma=(a_1\; a_2\;\cdots\; a_k)$$x\sigma x^{-1}=\Big(x(a_1)\;x(a_2)\;\cdots\; x(a_k)\Big)$, ie., a $k$-cycle is sent to another $k$-cycle
- $\frac{n!}{1^{n_1}n_1!.2^{n_2}n_2!\cdots n^{n_n}n_n!}:$ size of one conjugacy class
- $x=c_1c_2\cdots c_r\implies lcm\Big(o(c_1),o(c_2),\cdots,o(c_r)\Big)$
- $xy=yx$ $\&$ $\gcd(o(x),o(y))=1\implies o(xy)=o(x)o(y)$
- Transposition
- $(a_1\; a_2\;\cdots\; a_k)=(a_1\; a_k)(a_1\; a_{k-1})\cdots (a_1\; a_3)(a_1\; a_2)$
- $Alt(\Omega) / A_n:$ Alternating Group, $A_n \trianglelefteq S_n$, $[S_n:A_n]=2$
- $|G|=2n, n$ odd $\implies \exists H\trianglelefteq G$ s.t. $[G:H]=2$
- $A_n$ is simple for $n\ge 5$