We don't need much of the theory of finite groups. Here we summarize the concepts and ideas that we need, and fix some notation.

We of course need the definition of a group, of a subgroup and a normal subgroup, a homomorphism, and a quotient. We will consider subgroups of index 2 in Chapter 2, which are always normal. The second isomorphism theorem is that, whenever \(G\) is a group, \(H\) is a subgroup and \(K\) is a normal subgroup of \(G\), then \(HK/K\cong H/(H\cap K)\). The third isomorphism theorem states that, if \(G\) is a group and \(H\) and \(K\) are normal subgroups of \(G\) with \(K\leq H\), then \((G/K)/(H/K)\cong G/H\). Recall also that there is a one-to-one correspondence between subgroups of \(G\) containing a normal subgroup \(K\) of \(G\) and subgroups of \(G/K\) given by \(H\mapsto H/K\), and this bijection preserves normality.

The isomorphism theorems are needed when understanding the Jordan-Hölder theorem for modules, as they will be needed there in the module context. Recall that a simple group is one whose only normal subgroups are the identity and the whole group. A composition series is a chain of subgroups\[ 1=G_0\leq G_1\leq G_2\leq \cdots \leq G_n=G\]with \(G_{i-1}\) normal in \(G_i\) and \(G_i/G_{i-1}\) simple. The composition factors of \(G\) are the groups \(G_i/G_{i-1}\), and the Jordan-Hölder theorem states that the set of composition factors does not depend on the particular composition series one takes.

Permutation groups are used as a constant source of examples in the course, so the knowledge of what a group action is, when it is transitive, and so on, is of use.

We frequently use conjugacy. If \(x,g\in G\) then the conjugate, \(x^g\), is defined to be \(g^{-1}xg\). Extend this notation to subsets of \(G\), so \(X^g=\{x^g\mid x\in X\}\). The set conjugates of \(x\) is called the conjugacy class of \(x\), and denoted \(x^G\). If \(G\) is finite then \(|G|=|x^G|\cdot |C_G(x)|\), where \(C_G(x)\) is the centralizer of \(x\), all elements that commute with \(x\). The group \(G\) is the union of the conjugacy classes of \(G\). The normalizer of a subgroup \(H\), denoted \(N_G(H)\), is the set of all \(g\in G\) such that \(H^g=H\). The centre of \(G\) is \(Z(G)=C_G(G)\).

Endomorphisms are homomorphisms from a group to itself. Automorphisms are isomorphisms (bijective homomorphisms) from a group to itself. Conjugation by \(g\) is an automorphism of \(g\), denoted by \(c_g:G\to G\), \(c_g:x\mapsto x^g\). Automorphisms that are conjugation are called inner. The commutator \([x,y]\) is defined to be \(x^{-1}y^{-1}xy\), and the subgroup generated by them is called the derived subgroup, \(G'\). This is the smallest normal subgroup of \(G\) such that \(G/G'\) is abelian.

We need the direct product of two groups, \(G\) and \(H\), which is just the set of all pairs \((g,h)\) for \(g\in G\), \(h\in H\), with group multiplication being pointwise, so \((g,h)(g',h')=(gg',hh')\). One recognizes a group as a direct product by finding two trivially intersecting normal subgroups that generate the whole group.

If \(G\) is a finite \(p\)-group then \(Z(G)\neq 1\), and \(G/Z(G)\) cannot be cyclic. In particular, \(|G:Z(G)|\neq p\). If \(G\) is a finite group of order \(p^a\cdot m\) where \(p\not\mid m\) then Sylow's theorem states that there are subgroups of \(G\) of order \(p^a\), all such subgroups are conjugate, that the number of them is congruent to \(1\) modulo \(p\), and that any \(p\)-subgroup of \(G\) is contained in one of them. In particular, every group of order divisible by \(p\) contains elements of order \(p\). Elements of order \(2\) are called involutions.

A group is soluble if there is a chain of subgroups\[ 1=G_0\leq G_1\leq G_2\leq \cdots \leq G_n=G\]with \(G_{i-1}\) normal in \(G_i\) and \(G_i/G_{i-1}\) abelian. Subgroups and quotients of soluble groups are soluble, and if \(G/K\) and \(K\) are soluble then so is \(G\). A finite group is soluble if and only if it has no non-abelian simple composition factor.

In order to provide examples of representations we need some examples of groups. We will consider cyclic groups \(C_n\), the Klein four group, \(V_4=C_2\times C_2\), general finite abelian groups, which are direct products of cyclic groups, and dihedral groups \(D_{2n}\), the symmetries of a regular \(n\)-gon. The standard permutation groups we will use are the symmetric groups \(S_n\) and alternating groups \(A_n\). The last group that we see a lot is the quaternion group \(Q_8\). This is the group of fourth roots of unity of the quaternions \(\mathbb{H}\), so \(\{\pm1,\pm i,\pm j,\pm k\}\). This can be considered abstractly as the group of all products \(a^ib^j\), where \(a^4=1\), \(a^2=b^2\), and \(a^b=a^{-1}\).

(If you have not seen the quaternions \(\mathbb{H}\) before then we will construct them in the next section.)