Tour:Nonempty finite subsemigroup of group is subgroup
This article adapts material from the main article: subsemigroup of finite group is subgroup
This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Finite group| UP: Introduction two (beginners)| NEXT: Sufficiency of subgroup criterion
Expected time for this page: 8 minutes
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part
In a finite group, any nonempty subset that is closed under the operation of multiplication is, in fact, a subgroup. This isn't true for infinite groups (think for a moment about positive integers inside all integers).
WHAT YOU NEED TO DO: Understand the statement and proof below.
Statement
Verbal statement
Any nonempty multiplicatively closed finite subset (or equivalently, nonempty finite subsemigroup) of a group is a subgroup.
Symbolic statement
Let be a group and
be a nonempty finite subset such that
. Then,
is a subgroup of
.
Proof
Lemma
Statement of lemma: For any :
- All the positive powers of
are in
- There exists a positive integer
, dependent on
, such that
.
Proof: is closed under multiplication, so we get that the positive power of
are all in
. This proves (1).
Since is finite, the sequence
must have some repeated element. Thus, there are positive integers
such that
. Multiplying both sides by
, we get
. Set
, and we get
. Since
,
is a positive integer.
The proof
We prove that satisfies the three conditions for being a subgroup, i.e., it is closed under all the group operations:
- Binary operation: Closure under the binary operation is already given to us.
- Identity element
: Since
is nonempty, there exists some element
. Set
in the lemma. Applying part (2) of the lemma, we get that
is a positive power of
, so by part (1) of the lemma,
.
- Inverses
: Set
in the lemma. We make two cases:
- Case
: In this case
forcing
.
- Case
: In this case
is a positive power of
, hence by Part (1) of the lemma,
.
- Case
Related results
This page is part of the Groupprops guided tour for beginners. Make notes of any doubts, confusions or comments you have about this page before proceeding.
PREVIOUS: Finite group| UP: Introduction two (beginners)| NEXT: Sufficiency of subgroup criterion