Sufficiency of subgroup criterion
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This article gives the statement, and possibly proof, of a basic fact in group theory.
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This article gives a proof/explanation of the equivalence of multiple definitions for the term subgroup
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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.