Groups revisited

We recall the group theory of year one and sharpen it for what follows.

1. Generating sets (Def. 1.1) and the classification of cyclic groups (Thm 1.2) and their subgroups (Prop. 1.3).
2. Dihedral groups by generators and relations (Prop. 1.4); conjugacy in $S_n$ is cycle type (Prop. 1.5).
3. Direct products (Def. 1.6) and when $C_m\times C_n$ is cyclic (Prop. 1.7).
4. Cosets as equivalence classes and the index (Prop. 1.8).

For $S\subseteq G$, the subgroup generated by $S$, $\langle S\rangle$, is the intersection of all subgroups containing $S$; equivalently the set of finite products $s_1^{\pm1}s_2^{\pm1}\cdots s_k^{\pm1}$ with $s_i\in S$. $G$ is cyclic if $G = \langle g\rangle$ for some $g$.

Every cyclic group is isomorphic to $\mathbb{Z}$ (if infinite) or to $C_n = \mathbb{Z}/n\mathbb{Z}$ with $n = |G|$. In $C_n = \langle g\rangle$, $|g^k| = \frac{n}{\gcd(n, k)}$, so $g^k$ generates $C_n$ iff $\gcd(n, k) = 1$; there are $\varphi(n)$ generators.

Every subgroup of a cyclic group is cyclic. In $C_n$ there is exactly one subgroup of each order $d\mid n$, namely $\langle g^{n/d}\rangle$, and no others.

$D_n$ (symmetries of a regular $n$-gon, $n\ge3$) has order $2n$ and is generated by a rotation $r$ of order $n$ and a reflection $s$ with $srs^{-1} = r^{-1}$. Every element is uniquely $r^k$ or $r^ks$ ($0\le k < n$), and every $r^ks$ is a reflection (order $2$).

For $\sigma, \tau\in S_n$, $\tau\sigma\tau^{-1}$ is obtained from the cycle decomposition of $\sigma$ by replacing each entry $i$ by $\tau(i)$. Hence two permutations are conjugate iff they have the same cycle type, and the conjugacy classes of $S_n$ correspond to partitions of $n$.

The direct product $G\times H$ is the set of pairs with $(g, h)(g', h') = (gg', hh')$. The order of $(g, h)$ is $\operatorname{lcm}(|g|, |h|)$. A group $G$ is the internal direct product of subgroups $H, K$ if $H, K$ commute elementwise, $H\cap K = \{e\}$ and $HK = G$; then $G\cong H\times K$.

$C_m\times C_n\cong C_{mn}$ iff $\gcd(m, n) = 1$.

For $H\le G$, the relation $g\sim g'$ iff $g^{-1}g'\in H$ is an equivalence relation whose classes are the left cosets $gH$. All cosets have $|H|$ elements, and the number of left cosets equals the number of right cosets: the index $[G : H]$. For finite $G$, $|G| = [G : H]|H|$, and $[G : K] = [G : H][H : K]$ for $K\le H\le G$.

• Count elements of each order: cyclic groups have $\varphi(d)$ elements of order $d$.
• Abelian or not? $D_n$ ($n\ge3$) and $S_n$ ($n\ge3$) are not.
• Products: $C_m\times C_n$ is cyclic iff coprime.
• Use Lagrange to rule out subgroup orders, and cycle type for conjugacy in $S_n$.