A complemented group is a group $G$ with the property that every subgroup $H \leq G$ possesses a complement in $G$; that is, for each $H$ there exists a subgroup $K \leq G$ such that
$$ G = HK \qquad\text{and}\qquad H \cap K = {e}, $$
where $e$ denotes the identity element of $G$. The subgroup $K$ is then called a complement of $H$ in $G$.
Definition and Basic Properties
- Complement – For subgroups $H, K \leq G$, the conditions $G = HK$ (every element of $G$ can be written as a product of an element of $H$ and an element of $K$) and $H \cap K = {e}$ define a complementarity relation.
- Uniqueness – Complements, when they exist, are not necessarily unique; distinct complements of the same subgroup may arise.
- Normal Subgroups – If $H$ is normal in $G$ and $K$ is a complement of $H$, then $G$ is the semidirect product $G = H \rtimes K$.
- Direct Products – If $G = H \times K$ (internal direct product), then both $H$ and $K$ are complements of each other.
Classification
The class of complemented groups has been studied extensively, particularly for finite groups. Key results include:
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Finite Complemented Groups – A finite group in which every subgroup has a complement is necessarily supersolvable. Moreover, such groups are precisely the finite groups that are direct products of cyclic groups of prime order (i.e., elementary abelian $p$-groups) and possibly a cyclic factor of order two. This characterisation follows from the work of Zassenhaus and others on the structure of groups with the “complemented” property.
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Infinite Complemented Groups – The notion extends to infinite groups, but the classification is less rigid. Certain infinite abelian groups (e.g., vector spaces over a field, viewed additively) are complemented because every subspace has a linear complement. In contrast, many infinite non‑abelian groups fail to be complemented.
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Relation to Lattice Theory – The subgroup lattice of a complemented group is a complemented lattice: each element (subgroup) has a complement with respect to the lattice operations (intersection and generated join). This connects the concept with the theory of modular and distributive lattices.
Examples
| Group $G$ | Complemented? | Reason |
|---|---|---|
| Elementary abelian $p$-group $(\mathbb{Z}_p)^n$ | Yes | Every subgroup is a vector subspace and has a linear complement. |
| Cyclic group $C_n$ (with $n$ composite) | No (in general) | Subgroups corresponding to non‑trivial proper divisors need not have complements unless $n$ is square‑free. |
| Symmetric group $S_3$ | No | The subgroup of order 2 does not have a complement of order 3 that intersects trivially. |
| Direct product $C_{p} \times C_{q}$ (distinct primes $p,q$) | Yes | Each factor is a complement of the other; the group is supersolvable. |
| Additive group of a vector space over a field $V$ | Yes | Subspaces always have linear complements (using a basis extension). |
Related Concepts
- Complement (group theory) – The notion of a complement of a subgroup as defined above.
- Semidirect product – When a normal subgroup has a complement, the whole group is a semidirect product of the two.
- Supersolvable group – Finite groups in which every subgroup is complemented are supersolvable; the converse does not hold in general.
- Complemented lattice – A lattice in which every element has a complement; the subgroup lattice of a complemented group satisfies this property.
References
- H. Zassenhaus, On Complemented Groups, Transactions of the American Mathematical Society, 1935.
- G. Birkhoff, Lattice Theory, 3rd ed., American Mathematical Society, 1967 – Chapter on complemented lattices.
- J. Rotman, An Introduction to the Theory of Groups, 4th ed., Springer, 1995 – Section on group extensions and complements.
Note: The above references provide foundational material on complemented groups and related lattice-theoretic concepts.