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This article develops the basic language of groups, from binary
operations and subgroups through cosets, direct products, quotient
groups, and the three isomorphism theorems.
Definition. A law of composition, or
binary operation, on a set is a map , written or simply . It is associative if and commutative if
for all .
Definition. An identity element is an
element such that for every . If an identity exists, is invertible if there is
with ; then is denoted .
Example. Matrix multiplication on is associative and, for
, not commutative. Composition
in the symmetric group is
associative and, for , not
commutative.
Groups and subgroups
Definition. A group is a set with an associative binary operation,
an identity , and an inverse for
every element. It is abelian if its operation is commutative.
The order is the
cardinality of .
Example. The general linear group consists of the
invertible real matrices.
It is nonabelian for . The
special linear group
is a subgroup of .
Definition. A subset is a subgroup,
written , if it is a group
under the operation inherited from .
Theorem.Subgroup criterion. A
nonempty subset of a group is a subgroup if and only if for all .
Proposition. An arbitrary intersection of subgroups
of is a subgroup. The union of
two subgroups is a
subgroup if and only if or .
Subgroups of the integers
We write the additive group of integers simply as .
Theorem. Every subgroup of has the form for a unique integer .
Cyclic groups
Definition. A group is cyclic if for some . The order of is the least positive such that , if one exists, and is infinite
otherwise.
Theorem.Classification of cyclic
groups. Every finite cyclic group of order is isomorphic to , and every infinite
cyclic group is isomorphic to . In particular, every cyclic group is abelian.
Proof. The required isomorphisms are
and
◻
Theorem. For every prime , the multiplicative group is
cyclic.
Proof. Let and choose of maximal order . We
first show that for every
. Otherwise some prime occurs to a larger power in than in . Write and with and . Then has order , while has order . These elements commute and have
coprime orders, so their product has order , a contradiction. Thus every
element of is a root of . A polynomial of degree over the field has at most roots, so . Since , equality holds and . ◻
Theorem.Primitive root theorem.
The group is cyclic exactly when
where is an odd prime and
.
The converse statement “ cyclic implies prime” is false. For example, the group
is cyclic for .
Example. The smallest noncyclic group has order
: the Klein four-group
Homomorphisms
Definition. A homomorphism satisfies . Its kernel and
image are
A bijective homomorphism is an isomorphism; an isomorphism
is an
automorphism.
Proposition. If is a homomorphism, then
and ;
and ;
divides when ; equality holds if and only
if is injective on ;
is injective if and
only if .
Theorem.Cayley. Every group is isomorphic to a subgroup of the
symmetric group . If , then is isomorphic to a subgroup of .
Proof. The embedding is
◻
For , conjugation by
is the automorphism .
Equivalence relations and
partitions
Definition. An equivalence relation on is reflexive, symmetric, and
transitive. The equivalence class of is ; any element of a
class is a representative.
Theorem. The equivalence classes of an equivalence
relation form a partition of .
Conversely, every partition of
defines an equivalence relation by declaring two elements equivalent
exactly when they lie in the same part.
Every map therefore
defines an equivalence relation on by if and only if .
Cosets and Lagrange’s theorem
Definition. For , the left coset and right coset determined
by are
The sets of left and right cosets are denoted and , respectively. The
index is the number
of left cosets.
Proposition. Left cosets partition , every left coset is in bijection with
, and
Similarly, if and only if
.
Theorem.Lagrange. If is finite and , then . In particular, divides .
Corollary. If
is finite, the order of every divides .
Consequently, a group of prime order is cyclic, generated by any
nonidentity element.
Definition. A subgroup is normal, written , if for every .
Proposition. For , the following are equivalent:
;
for every ;
every left coset of is
also a right coset of .
Proof. For , given , condition (3) gives for some . Since , write with . Then , so , and hence . ◻
Modular arithmetic
For , define when . Its equivalence classes are
the congruence classes modulo .
The number of units of is Euler’s totient
The correspondence theorem
Theorem.Correspondence theorem for
groups. Let be a homomorphism. The map is an
inclusion-preserving bijection between the subgroups of containing and the subgroups of . Its inverse sends
to . Normal subgroups correspond
to normal subgroups, and
whenever these quantities are finite.
Proof. The mutually inverse correspondences are
with and
on the stated
domains. This is a lattice isomorphism between subgroup lattices, not in
general a group isomorphism ; rather, . ◻
Product groups
Definition. The direct product has componentwise
multiplication:
Theorem.Internal direct product
criterion. Suppose , the
subgroups generate , and
Then the multiplication map
is an isomorphism. In this situation is the internal direct product of the
.
Proof. Let ,
, and . Normality of both factors
gives
Therefore :
elements belonging to distinct factors commute. ◻
Quotient groups and
isomorphism theorems
Definition. If , the quotient
group is the set of cosets
with multiplication .
Normality is precisely what makes this operation well-defined. The
natural projection ,
, is surjective with
kernel .
Theorem.First isomorphism theorem.
For a homomorphism ,
Proof. The isomorphism is
◻
Theorem.Second isomorphism
theorem. If and
, then , , and
Proof. The homomorphism and its induced isomorphism are
◻
Theorem.Third isomorphism theorem.
If , , and , then and
Proof. The required homomorphism and its induced isomorphism
are