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## Saturday, October 31, 2009

### When closure is sufficient for a subset to be a subgroup.

While browsing Group Theory I ( Suzuki ) I noticed the following proposition.

If G is a finite group and S is a subset of G then closure in S suffices for S to be a subgroup.

Proof:
S is a subgroup if for all a,b,c in S
( i ) ab in S - closure
( ii ) a(bc)= (ab)c - associativity
( iii ) e in S - has identity
( iv ) a^(-1) in S - has inverse
let's prove them one by one:
( i) is proposed to be true ;
( ii ) is true for G and thus true for S ;
( iii ) since G is finite there is an integer n such that a^n = e thus e in S
( iv) since a a^(n-1) = e all a have an inverse.
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When in exercises the word 'finite' is added to group like 'G is a finite group ' we know that for G the group axioms are true like (i) to (iv) above AND that there is an integer n such that for all g in G g^n = e.

## Welcome to The Bridge

Mathematics: is it the fabric of MEST?
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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)