In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, more specifically
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, the three subgroups lemma is a result concerning
commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
s. It is a consequence of
Philip Hall
Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups.
Biography
He was educated first at Christ's Hospital, where he won the Thomps ...
and
Ernst Witt's
eponymous identity.
Notation
In what follows, the following notation will be employed:
* If ''H'' and ''K'' are
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
s of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''G'', the commutator of ''H'' and ''K'', denoted by
'H'', ''K'' is defined as the subgroup of ''G'' generated by
commutators
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, ...
between elements in the two subgroups. If ''L'' is a third subgroup, the convention that
'H'',''K'',''L''=
''H'',''K''''L''] will be followed.
* If ''x'' and ''y'' are elements of a group ''G'', the
Conjugate (group theory), conjugate of ''x'' by ''y'' will be denoted by
.
* If ''H'' is a subgroup of a group ''G'', then the
centralizer
In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...
of ''H'' in ''G'' will be denoted by C
G(''H'').
Statement
Let ''X'', ''Y'' and ''Z'' be subgroups of a group ''G'', and assume
:
and
Then
.
More generally, for a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of
, if
and
, then
.
[Isaacs, Corollary 8.28, p. 111]
Proof and the Hall–Witt identity
Hall–Witt identity
If
, then
:
Proof of the three subgroups lemma
Let
,
, and
. Then