finite subgroup

Theorem.  A non-empty finite subset K of a group G is a subgroupMathworldPlanetmathPlanetmath of G if and only if

xyKfor allx,yK. (1)

Proof.  The condition (1) is apparently true if K is a subgroup.  Conversely, suppose that a nonempty finite subset K of the group G satisfies (1).  Let a and b be arbitrary elements of K.  By (1), all () powers of b belong to K.  Because of the finiteness of K, there exist positive integers r,s such that


By (1),


Thus also  ab-1K,  whence, by the theorem of the entry, K is a subgroup of G.

Example.  The multiplicative groupMathworldPlanetmath G of all nonzero complex numbersMathworldPlanetmathPlanetmath has the finite multiplicative subset{1,-1,i,-i},  which has to be a subgroup of G.

Title finite subgroup
Canonical name FiniteSubgroup
Date of creation 2013-03-22 18:57:02
Last modified on 2013-03-22 18:57:02
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Theorem
Classification msc 20A05
Synonym criterion for finite subgroup
Synonym finite subgroup criterion