10.4 Some applications of Lagrange’s theorem
Lagrange’s Theorem gives most information about a group when the order of the group has relatively few factors, as then it puts more restrictions on possible orders of subgroups and elements.
Let’s consider the extreme case, when the order of the group is a prime \(p\), and so the only factors are \(1\) and \(p\).
Let \(p\) be a prime and \(G\) a group with \(|G|=p\). Then
\(G\) is cyclic.
Every element of \(G\) except the identity has order \(p\) and generates \(G\).
The only subgroups of \(G\) are the trivial subgroup \(\{e\}\) and \(G\) itself.
Let \(x\in G\) with \(x\neq e\). Then \(\operatorname{ord}(x)\) divides \(|G|=p\) by Corollary 10.20, and \(\operatorname{ord}(x)\neq1\) since \(x\neq e\). So \(\operatorname{ord}(x)=p\). So the cyclic subgroup \(\langle x\rangle\) generated by \(x\) has order \(p\), and so must be the whole of \(G\). This proves a.) and b.).
Let \(H\leq G\). Then by Theorem 10.15 \(|H|\) divides \(|G|=p\), and so \(|H|=1\), in which case \(H\) is the trivial subgroup \(\{e\}\), or \(|H|=p\), in which case \(H=G\).□
If \(p\) is prime and \(P\) and \(Q\) are two subgroups of a group \(G\) with \(|P|=p=|Q|\), then either \(P=Q\) or \(P\cap Q=\{e\}\).
If \(P\cap Q\neq\{e\}\) then choose \(x\in P\cap Q\) with \(x\neq e\). By the previous theorem, \(x\) generates both \(P\) and \(Q\), so \(P=\langle x\rangle=Q\).
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Now some other simple general consequences of Lagrange’s Theorem.
Let \(G\) be a group and \(H\),\(K\) two finite subgroups of \(G\) with \(|H|=m\), \(|K|=n\) and \(\gcd(m,n)=1\). Then \(H\cap K=\{e\}\).
Recall that the intersection of two subgroups is itself a subgroup, so that \(I=H\cap K\) is a subgroup both of \(H\) and of \(K\). Since it’s a subgroup of \(H\), Lagrange’s Theorem implies that \(|I|\) divides \(m=|H|\). But similarly \(|I|\) divides \(n=|K|\). So since \(\gcd(m,n)=1\), \(|I|=1\) and so \(I=\{e\}\).
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Let \(G\) be a group with \(|G|=4\). Then either \(G\) is cyclic or \(G\) is isomorphic to the Klein \(4\)-group \(C_2\times C_2\). In particular there are just two non-isomorphic groups of order \(4\), both abelian.
By Corollary 10.20 the order of any element of \(G\) divides \(4\), and so must be \(1\), \(2\) or \(4\).
If \(G\) has an element of order \(4\) then it is cyclic.
If not, it must have one element (the identity \(e\)) of order \(1\) and three elements \(a,b,c\) of order \(2\). So \(a^{-1}=a\), \(b^{-1}=b\) and \(c^{-1}=c\).
Consider which element is \(ab\). If \(ab=e\) then \(b=a^{-1}\), which is false, since \(a^{-1}=a\). If \(ab=a\) then \(b=e\), which is also false. If \(ab=b\) then \(a=e\), which is also false. So \(ab=c\).
Similarly \(ba=c\), \(ac=b=ca\) and \(bc=a=cb\), and \(G\) is isomorphic to the Klein \(4\)-group.□
We’ll finish this section with some other results about groups of small order that we won’t prove. These are easier to prove with a bit more theory, which are all proved in the third year Group Theory unit.
Let \(p\) be an odd prime. Then every group of order \(2p\) is either cyclic or isomorphic to the dihedral group \(D_{2p}\).
Let \(p\) be a prime. Every group of order \(p^2\) is either cyclic or isomorphic to \(C_P\times C_p\) (and so in particular is abelian).
However there are non-abelian groups of order \(p^3\) for every prime \(p\). The dihedral group \(D_8\) is one example for \(p=2\).
There are five groups of order \(8\) up to isomorphism. Three, \(C_8\), \(C_4\times C_2\) and \(C_2\times C_2\times C_2\), are abelian, and two, the dihedral group \(D_8\) and another group \(Q_8\) called the quaternion group are non-abelian.
The first few orders not dealt with by the general theorems above are \(12\), \(15\), \(16\) and \(18\). It turns out that are five non-isomorphic groups of order \(12\), every group of order \(15\) is cyclic, there are fourteen non-isomorphic groups of order \(16\), and five of order \(18\).
The number of non-isomorphic groups of order \(2^n\) grows very quickly with \(n\). There are \(49,487,365,422\) (nearly fifty billion) non-isomorphic groups of order \(1024=2^{10}\).