Subgroup
A non-empty subset of a group
which itself is a group with respect to the operation defined on
. A subset
of a group
is a subgroup of
if and only if: 1)
contains the product of any two elements from
; and 2)
contains together with any element
the inverse
. In the cases of finite and periodic groups, condition 2) is superfluous.
The subset of a group G consisting of the element 1 only is clearly a subgroup; it is called the unit subgroup of and is usually denoted by
. Also,
itself is a subgroup. A subgroup different from
is called a proper subgroup of
. A proper subgroup of an infinite group can be isomorphic to the group itself. The group
itself and the subgroup
are called improper subgroups of
, while all the others are called proper ones.
The set-theoretic intersection of any two (or any set of) subgroups of a group is a subgroup of
. The intersection of all subgroups of
containing all elements of a certain non-empty set
is called the subgroup generated by the set
and is denoted by
. If
consists of one element
, then
is called the cyclic subgroup of the element
. A group that coincides with one of its cyclic subgroups is called a cyclic group.
A set-theoretic union of subgroups is, in general, not a subgroup. By the join of subgroups ,
, one means the subgroup generated by the union of the sets
.
The product of two subsets and
of a group
is the set consisting of all possible (different) products
, where
,
. The product of two subgroups
is a subgroup if and only if
, and in that case the product
coincides with the subgroup generated by
and
(i.e. with the join of
and
).
A homomorphic image of a subgroup is a subgroup. If a group is isomorphic to a subgroup
of a group
, one says that
can be imbedded in
(as groups). If one is given two groups and each of them is isomorphic to a proper subgroup of the other, it does not necessarily follow that these groups themselves are isomorphic (cf. also Homomorphism; Isomorphism).
Comments
References
[a1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |
[a2] | M. Hall jr., "The theory of groups" , Macmillan (1959) pp. 124 |
Subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subgroup&oldid=33209