Tour:Sufficiency of subgroup criterion
This article adapts material from the main article: sufficiency of subgroup criterion
This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Subsemigroup of finite group is subgroup| UP: Introduction two (beginners)| NEXT: Manipulating equations in groups
Expected time for this page: 8 minutes
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part
WHAT YOU NEED TO DO:
- Recall the various definitions of subgroup, including the subgroup criterion
- Try proving that the subgroup criterion is necessary and sufficient
- Read below the proof of sufficiency
- Compare with the result we just saw for finite groups (any nonempty multiplicatively closed subset is a subgroup)
PONDER:
- What axioms of group structure play a role in this proof?
- In what ways are things different for finite groups, and why?
Statement
For a subset of a group
, the following are equivalent:
-
is a subgroup, viz
is closed under the binary operation of multiplication, the inverse map, and contains the identity element
-
is a nonempty set closed under left quotient of elements (that is, for any
in
,
is also in
)
-
is a nonempty set closed under right quotient of elements (that is, for any
in
,
is also in
)
Proof
We shall here prove the equivalence of the first two conditions. Equivalence of the first and third conditions follows by analogous reasoning.
(1) implies (2)
Clearly, if is a subgroup:
-
is nonempty since
contains the identity element
- Whenever
are in
so is
and hence
(2) implies (1)
Suppose is a nonempty subset closed under left quotient of elements. Then, pick an element
from
. (VIDEO WARNING: In the embeddded video, the letter
is used in place of
, which is a little unwise, but the spirit of reasoning is the same).
-
is in
: Set
to get
is contained in
, hence
is in
-
: Now that
is in
, set
to get
is also in
, so
is in
-
: Set
. The previous step tells us both are in
. So
is in
, which tells us that
is in
.
Thus, satisfies all the three conditions to be a subgroup.
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: Subsemigroup of finite group is subgroup| UP: Introduction two (beginners)| NEXT: Manipulating equations in groups