MTH3003 Lecture 18

The final lecture of the course shifts gears: rather than developing more general theory, we classify finite abelian groups completely. The key step is showing that the internal direct product of two normal subgroups with trivial intersection is isomorphic to their external direct product. This unlocks the Fundamental Theorem of Finite Abelian Groups: every such group is a direct product of cyclic prime-power groups.

External vs Internal Direct Products

Recap: External Direct Product

Given groups $G$ and $H$, the external direct product $G\times H$ is the set $\{(g,h):g\in G,h\in H\}$ with componentwise multiplication

$$(g_1,h_1)\cdot (g_2,h_2)=(g_1{g}_2,h_1{h}_2).$$

For finite groups, $|G\times H|=|G|\cdot |H|$.

External direct products are easy to construct (“here are two groups, take their pairs”) but hard to detect when they appear inside another group. We now develop a tool to detect them.

Note

Proposition. $G\times H\cong H\times G$. (Swap coordinates.)

Bridge: Lemma on Commuting Normal Subgroups

Lemma

Suppose $M,N⊴G$ with $M\cap N=\{e\}$. Then for all $m\in M$ and $n\in N$,

$$mn=nm.$$

Elements of two normal subgroups with trivial intersection commute, even if $G$ is non-abelian!

Proof. Consider the commutator $n^{-1}mn{m}^{-1}$. We show it lies in both $M$ and $N$, hence in $M\cap N=\{e\}$.

• $n^{-1}mn\in M$ because $M$ is normal. So $n^{-1}mn{m}^{-1}=(n^{-1}mn)\cdot m^{-1}\in M$.
• $mn{m}^{-1}\in N$ because $N$ is normal. So $n^{-1}mn{m}^{-1}=n^{-1}\cdot (mn{m}^{-1})\in N$.

Hence $n^{-1}mn{m}^{-1}\in M\cap N=\{e\}$, giving $n^{-1}mn{m}^{-1}=e$, i.e. $mn=nm$. $■$

Theorem: Detecting Direct Products Internally

Theorem

Suppose $M,N⊴G$ with $M\cap N=\{e\}$. Then $MN\le G$ and

$$\boxed{MN\cong M\times N.}$$

Proof sketch. Define $\theta :MN\to M\times N$ by $\theta (mn)=(m,n)$.

• Well-defined. If $g=mn=m_1{n}_1$ for $m,m_1\in M$, $n,n_1\in N$, then $m^{-1}{m}_1=n{n}_1^{-1}\in M\cap N=\{e\}$, so $m=m_1$ and $n=n_1$. The decomposition is unique.
• Homomorphism. $\theta (m_1{n}_1)\theta (m_2{n}_2)=(m_1,n_1)(m_2,n_2)=(m_1{m}_2,n_1{n}_2)$. By the lemma, $n_1{m}_2=m_2{n}_1$, so $m_1{n}_1{m}_2{n}_2=m_1{m}_2{n}_1{n}_2$, and $\theta (m_1{n}_1{m}_2{n}_2)=(m_1{m}_2,n_1{n}_2)$ - matches.
• Injective. $\theta (m_1{n}_1)=\theta (m_2{n}_2)\Rightarrow (m_1,n_1)=(m_2,n_2)\Rightarrow m_1=m_2,n_1=n_2$.
• Surjective. Any $(m,n)\in M\times N$ has $\theta (mn)=(m,n)$.

Hence $\theta$ is an isomorphism. $■$

Definition

If $G$ has normal subgroups $M,N$ with $M\cap N=\{e\}$ and $G=MN$, then $G$ is the internal direct product of $M$ and $N$.

The theorem says: internal direct products are isomorphic to external direct products. So once we spot the internal pattern (two normal subgroups with trivial intersection whose product is everything), we conclude $G\cong M\times N$.

Generalises immediately

The same construction extends to any finite number of normal subgroups $M_1,\dots ,M_k⊴G$ with pairwise trivial intersections (and the right product condition): $G\cong M_1\times M_2\times \cdots \times M_k$.

Applying It: Order 15 Revisited

In lecture 17 we proved every group of order 15 is cyclic by counting elements. Here is a slicker proof using the internal-direct-product theorem.

Example

Let $|G|=15$. Then $G\cong C_3\times C_5\cong C_{15}$.

Sketch. $|G|=3\cdot 5$. By Sylow’s theorems, $G$ has unique (hence normal) Sylow $3$-subgroup $J$ (order $3$) and unique (hence normal) Sylow $5$-subgroup $K$ (order $5$). $J\cap K=\{e\}$ by Lagrange ($|J\cap K|$ divides both $3$ and $5$, so equals $1$). Hence $JK\cong J\times K\cong C_3\times C_5$. By Lagrange $|JK|=|J|\cdot |K|/|J\cap K|=15=|G|$, so $JK=G$. Thus $G\cong C_3\times C_5\cong C_{15}$ (the second isomorphism uses $\gcd (3,5)=1$). $■$

Note

This pattern - “Sylow subgroups are unique, hence normal, hence $G$ decomposes as their direct product” - is the workhorse of small-order group classification.

The Fundamental Theorem of Finite Abelian Groups

For abelian groups the entire subgroup lattice cooperates: every subgroup is normal, so the conditions of the internal-direct-product theorem are automatically satisfied.

Fundamental Theorem of Finite Abelian Groups

Every finite abelian group $G$ is isomorphic to a direct product of cyclic groups of prime-power order:

$$G\cong C_{p_1^{r_1}}\times C_{p_2^{r_2}}\times \cdots \times C_{p_k^{r_k}},$$

where the $p_i$ are (not necessarily distinct) primes. The decomposition is unique up to reordering.

In particular, the only finite abelian groups of order $n$ are determined by the partitions of the multiplicities in the prime factorisation of $n$.

Examples

| $|G|$ | Factorisation | Abelian groups, up to isomorphism | |---|---|---| | 4 | $2^2$ | $C_4$, $C_2\times C_2$ | | 8 | $2^3$ | $C_8$, $C_4\times C_2$, $C_2\times C_2\times C_2$ | | 9 | $3^2$ | $C_9$, $C_3\times C_3$ | | 12 | $2^2\cdot 3$ | $C_4\times C_3\cong C_{12}$, $C_2\times C_2\times C_3\cong C_2\times C_6$ | | 16 | $2^4$ | $C_{16}$, $C_8\times C_2$, $C_4\times C_4$, $C_4\times C_2\times C_2$, $C_2\times C_2\times C_2\times C_2$ |

The number of abelian groups of order $p^n$ equals the number of integer partitions of $n$.

Note

The Fundamental Theorem builds on Sylow theory: an abelian $G$ decomposes as the direct product of its (unique, since abelian!) Sylow $p$-subgroups, then each Sylow $p$-subgroup itself decomposes as a direct product of cyclic $p$-groups via more refined arguments.

Course Synthesis

Looking back over the 18 lectures, the course has built up the following ladder:

1. Permutations (lectures 1-2): cycle notation, products, signature.
2. Groups and subgroups (3-6): axioms, generators, cyclic groups, dihedral groups, isometry groups.
3. Cosets and Lagrange (7-8): partition of $G$, divisibility constraints, Cauchy’s theorem.
4. Quotients, homomorphisms, isomorphism theorems (9-11): structural maps, kernels, the three isomorphism theorems.
5. Alternating groups and signature (12-13): even/odd permutations, $A_n$.
6. Group actions (14): generalisation of the symmetric-group viewpoint, Cayley’s theorem.
7. Orbits and stabilisers (15): orbit-stabiliser theorem.
8. Burnside / counting (16): orbit counting theorem, applications to colouring problems.
9. Sylow theory (17): existence, conjugacy, and counting of $p$-Sylow subgroups.
10. Direct products and abelian classification (18): internal-direct-product detection, fundamental theorem of finite abelian groups.

The driving theme: groups are most usefully studied through their actions on sets, and through their decomposition into well-understood pieces (cyclic, abelian, simple).

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Pre-Lecture Notes from mth3003 lecture notes 18.pdf

• External direct product $G\times H$: pairs with componentwise multiplication; $|G\times H|=|G|\cdot |H|$.
• Lemma: if $M,N⊴G$ with $M\cap N=\{e\}$, then $mn=nm$ for all $m\in M$, $n\in N$. Proof via the commutator $n^{-1}mn{m}^{-1}\in M\cap N=\{e\}$.
• Internal direct product theorem: $M,N⊴G$ with $M\cap N=\{e\}\Rightarrow MN\le G$ and $MN\cong M\times N$. Detect direct products by spotting two normal subgroups with trivial intersection.
• Definition: $G$ is the internal direct product of $M$ and $N$ if $G=MN$, $M\cap N=\{e\}$, and $M,N⊴G$.
• Generalisation: applies to any number of normal subgroups with pairwise trivial intersections.
• Application: re-prove $|G|=15\Rightarrow G\cong C_{15}$ by recognising $G\cong C_3\times C_5$.
• Fundamental Theorem of Finite Abelian Groups: every finite abelian group is a direct product of cyclic prime-power groups, uniquely (up to order). Number of abelian groups of order $p^n$ = number of partitions of $n$.
• End of course: see the mth3003 final exam cheat sheet for a complete formula reference.