1.3: Subgroups

[bd4] Let \(S\) be a nonempty subset of a group \(G\). If

S1:

\(a,b\in S\implies ab\in S\), and

S2:

\(a\in S\implies a^{-1}\in S,\)

then the binary operation on \(G\) makes \(S\) into a group.

(S1) implies that the binary operation on \(G\) defines a binary operation \(S\times S\rightarrow S\) on \(S\), which is automatically associative. By assumption \(S\) contains at least one element \(a\), its inverse \(a^{-1}\), and the product \(e=aa^{-1}\). Finally (S2) shows that the inverses of elements in \(S\) lie in \(S\).

A nonempty subset \(S\) satisfying (S1) and (S2) is called a subgroup of \(G\). When \(S\) is finite, condition (S1) implies (S2): let \(a\in S\); then \(\{a,a^{2},\ldots\}\subset S\), and so \(a\) has finite order, say \(a^{n}=e\); now \(a^{-1}=a^{n-1}\in S\). The example \((\mathbb{N},+)\subset(\mathbb{Z},+)\) shows that (S1) does not imply (S2) when \(S\) is infinite.

[bd4e]The centre of a group \(G\) is the subset

\[Z(G)=\{g\in G\mid gx=xg\text{ for all }x\in G\}. \nonumber \]

It is a subgroup of \(G\).

[bd5] An intersection of subgroups of \(G\) is a subgroup of \(G.\)

It is nonempty because it contains \(e\), and (S1) and (S2) obviously hold.

[bd6] It is generally true that an intersection of subobjects of an algebraic object is a subobject. For example, an intersection of subrings of a ring is a subring, an intersection of submodules of a module is a submodule, and so on.

[bd7] For any subset \(X\) of a group \(G\), there is a smallest subgroup of \(G\) containing \(X\). It consists of all finite products of elements of \(X\) and their inverses (repetitions allowed).

The intersection \(S\) of all subgroups of \(G\) containing \(X\) is again a subgroup containing \(X\), and it is evidently the smallest such group. Clearly \(S\) contains with \(X\), all finite products of elements of \(X\) and their inverses. But the set of such products satisfies (S1) and (S2) and hence is a subgroup containing \(X\). It therefore equals \(S\).

The subgroup \(S\) given by the proposition is denoted \(\langle X\rangle\), and is called the subgroup generated by \(X\). For example, \(\langle\emptyset\rangle=\{e\}\). If every element of \(X\) has finite order, for example, if \(G\) is finite, then the set of all finite products of elements of \(X\) is already a group and so equals \(\langle X\rangle\).

We say that \(X\) generates \(G\) if \(G=\langle X\rangle\), i.e., if every element of \(G\) can be written as a finite product of elements from \(X\) and their inverses. Note that the order of an element \(a\) of a group is the order of the subgroup \(\langle a\rangle\) it generates.

Examples

[bd8a]The cyclic groups. A group is said to be cyclic if it is generated by a single element, i.e., if \(G=\langle r\rangle\) for some \(r\in G\). If \(r\) has finite order \(n\), then

\[G=\{e,r,r^{2},...,r^{n-1}\}\approx C_{n},\quad r^{i}\leftrightarrow i\mod n, \nonumber \]

and \(G\) can be thought of as the group of rotational symmetries about the centre of a regular polygon with \(n\)-sides. If \(r\) has infinite order, then

\[G=\{\ldots,r^{-}{}^{i},\ldots,r^{-1},e,r,\ldots,r^{i},\ldots\}\approx C_{\infty},\quad r^{i}\leftrightarrow i. \nonumber \]

Thus, up to isomorphism, there is exactly one cyclic group of order \(n\) for each \(n\leq\infty\). In future, we shall loosely use \(C_{n}\) to denote any cyclic group of order \(n\) (not necessarily \(\mathbb{Z}{}/n\mathbb{Z}{}\) or \(\mathbb{Z}{}\)).

[bd8b]The dihedral groups \(D_{n}\).⁴ For \(n\geq3\), \(D_{n}\) is the group of symmetries of a regular polygon with \(n\)-sides.⁵ Number the vertices \(1,\ldots,n\) in the counterclockwise direction. Let \(r\) be the rotation through \(2\pi/n\) about the centre of polygon (so \(i\mapsto i+1\mod n)\), and let \(s\) be the reflection in the line (= rotation about the line) through the vertex \(1\) and the centre of the polygon (so \(i\mapsto n+2-i\mod n\)). For example, the pictures

\[\begin{aligned} s & =\left\{ \begin{array} [c]{c}% 1\leftrightarrow1\\ 2\leftrightarrow3 \end{array} \right. \\ r & =1\rightarrow2\rightarrow3\rightarrow1\end{aligned} \nonumber \]

\[\begin{aligned} s & =\left\{ \begin{array} [c]{c}% 1\leftrightarrow1\\ 2\leftrightarrow4\\ 3\leftrightarrow3 \end{array} \right. \\ r & =1\rightarrow2\rightarrow3\rightarrow4\rightarrow1\end{aligned} \nonumber \]

illustrate the groups \(D_{3}\) and \(D_{4}\). In the general case

\[r^{n}=e;\quad s^{2}=e;\quad srs=r^{-1}\quad\text{(so }sr=r^{n-1}s). \nonumber \]

These equalites imply that

\[D_{n}=\{e,r,...,r^{n-1},s,rs,...,r^{n-1}s\}, \nonumber \]

and it is clear from the geometry that the elements of the set are distinct, and so \(|D_{n}|=2n\).

Let \(t\) be the reflection in the line through the midpoint of the side joining the vertices \(1\) and \(2\) and the centre of the polygon (so \(i\mapsto n+3-i\mod n)\). Then \(r=ts\), because

\[i\overset{s}{\mapsto}n+2-i\overset{t}{\mapsto}n+3-(n+2-i)=i+1\mod n. \nonumber \]

Hence \(D_{n}=\langle s,t\rangle\) and

\[s^{2}=e,\quad t^{2}=e,\quad(ts)^{n}=e=(st)^{n}. \nonumber \]

We define \(D_{1}\) to be \(C_{2}=\{1,r\}\) and \(D_{2}\) to be \(C_{2}\times C_{2}=\{1,r,s,rs\}\). The group \(D_{2}\) is also called the Klein Vierergruppe or, more simply, the 4-group and denoted \(V\) or \(V_{4}\). Note that \(D_{3}\) is the full group of permutations of \(\{1,2,3\}\). It is the smallest noncommutative group.

By adding a tick at each vertex of a regular polygon, we can reduce its symmetry group from \(D_{n}\) to \(C_{n}\). By adding a line from the centre of the polygon to the vertex \(1\), we reduce its symmetry group to \(\langle s\rangle\). Physicist like to say that we have “broken the symmetry”.

[bd8c]The quaternion group \(Q\): Let \(a=\left( \begin{smallmatrix} 0 & \sqrt{-1}\\ \sqrt{-1} & 0 \end{smallmatrix} \right)\) and \(b=\left( \begin{smallmatrix} \hfill0 & 1\\ -1 & 0 \end{smallmatrix} \right)\). Then

\[a^{4}=e,\quad a^{2}=b^{2},\quad bab^{-1}=a^{3}\text{ (so }ba=a^{3}b\text{).}% \nonumber \]

The subgroup of \(\GL_{2}(\mathbb{C})\) generated by \(a\) and \(b\) is

\[Q=\{e,a,a^{2},a^{3},b,ab,a^{2}b,a^{3}b\}. \nonumber \]

The group \(Q\) can also be described as the subset \(\{\pm1,\pm i,\pm j,\pm k\}\) of the quaternion algebra \(\mathbb{H}{}\). Recall that

\[\mathbb{H}{}=\mathbb{R}{}1\oplus\mathbb{R}{}i\oplus\mathbb{R}{}j\oplus \mathbb{R}{}k \nonumber \]

with the multiplication determined by

\[i^{2}=-1=j^{2},\quad ij=k=-ji.\quad \nonumber \]

The map \(i\mapsto a\), \(j\mapsto b\) extends uniquely to a homomorphism \(\mathbb{H}{}\rightarrow M_{2}(\mathbb{C}{})\) of \(\mathbb{R}{}\)-algebras, which maps the group \(\langle i,j\rangle\) isomorphically onto \(\langle a,b\rangle\).

[bd8d]Recall that \(S_{n}\) is the permutation group on \(\{1,2,...,n\}\). A transposition is a permutation that interchanges two elements and leaves all other elements unchanged. It is not difficult to see that \(S_{n}\) is generated by transpositions (see ([ga21]) below for a more precise statement).