Sufficiency of subgroup criterion
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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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 )
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.