Group theory starts by isolating the algebra of symmetry. The point is not only to list axioms, but to learn how subgroups, generated subgroups, and homomorphisms let one compare different symmetry systems.
Group axioms and first examples
Definition. A group is a set \(G\) with a binary operation \((g,h)\mapsto gh\) such that multiplication is associative, there is an identity element \(e\), and every \(g\in G\) has an inverse \(g^{-1}\). The group is abelian if \(gh=hg\) for all \(g,h\in G\).
Example. The additive group \((\mathbb Z,+)\), the multiplicative group \(F^\times\) of a field, the symmetric group \(S_n\), and the dihedral group \(D_{2n}\) of symmetries of a regular \(n\)-gon are basic examples. The first two are abelian; \(S_n\) is nonabelian for \(n\ge 3\).
Counterexample. The set \(M_n(F)\) of all matrices under multiplication is not a group, because singular matrices have no inverses. The invertible matrices form the group \(\operatorname{GL}_n(F)\).
Subgroups and subgroup tests
Definition. A subset \(H\subset G\) is a subgroup, written \(H\le G\), if \(H\) is a group under the operation inherited from \(G\). Equivalently, \(H\ne\varnothing\) and \(ab^{-1}\in H\) for all \(a,b\in H\).
Proof. If \(H\) is a subgroup, then \(ab^{-1}\in H\). Conversely, choose \(a\in H\). Then \(e=aa^{-1}\in H\), and \(a^{-1}=ea^{-1}\in H\). Finally \(ab=a(b^{-1})^{-1}\in H\). Associativity is inherited from \(G\).
Example. In \(\mathbb Z\), every subgroup is \(n\mathbb Z\) for a unique \(n\ge0\). If \(H\le\mathbb Z\) is nonzero, choose its smallest positive element \(n\). Division with remainder shows every element of \(H\) is a multiple of \(n\).
Generated subgroups and cyclic groups
Definition. If \(S\subset G\), the subgroup generated by \(S\), written \(\langle S\rangle\), is the smallest subgroup of \(G\) containing \(S\). For one element \(g\), \(\langle g\rangle=\{g^n:n\in\mathbb Z\}\). A group is cyclic if \(G=\langle g\rangle\) for some \(g\).
Proposition. Every subgroup of a cyclic group is cyclic. If \(G=\langle g\rangle\) has order \(n\), then for every divisor \(d\mid n\) there is exactly one subgroup of order \(d\), namely \(\langle g^{n/d}\rangle\).
Proof. If \(H\le\langle g\rangle\), choose the smallest positive integer \(m\) with \(g^m\in H\), unless \(H=\{e\}\). Division of exponents by \(m\) shows every \(g^k\in H\) is a power of \(g^m\). The finite statement follows by computing the order of \(g^r\), which is \(n/\gcd(n,r)\).
Homomorphisms, kernels, and images
Definition. A homomorphism \(\varphi:G\to H\) satisfies \(\varphi(gh)=\varphi(g)\varphi(h)\). Its kernel and image are \[ \ker\varphi=\{g\in G:\varphi(g)=e_H\},\qquad \operatorname{im}\varphi=\{\varphi(g):g\in G\}. \]
Proposition. The kernel is a normal subgroup of \(G\), the image is a subgroup of \(H\), and \(\varphi\) is injective exactly when \(\ker\varphi=\{e_G\}\).
Proof. The subgroup assertions follow from the subgroup test. For normality, if \(k\in\ker\varphi\), then \(\varphi(gkg^{-1})=\varphi(g)e_H\varphi(g)^{-1}=e_H\). Finally \(\varphi(g)=\varphi(h)\) iff \(h^{-1}g\in\ker\varphi\).
Orders and arithmetic restrictions
Definition. The order \(|g|\) of \(g\in G\) is the order of \(\langle g\rangle\). If no positive power of \(g\) is the identity, then \(g\) has infinite order.
Corollary. If \(G\) is finite, then \(|g|\) divides \(|G|\) for every \(g\in G\), and \(g^{|G|}=e\).
Remark. The order of an element is often the first invariant to compute. In \(S_n\), the order of a permutation is the least common multiple of the lengths of its disjoint cycles.