Finite Groups, The Sum of the Elements in an Abelian Group
The Sum of the Elements in an Abelian Group
Let G be an abelian group,
and add up all the elements in G.
Pair each element with its inverse.
These cancel, leaving only the involutions.
The subgroup of involutions is
Z2k,
as each involution is present or not.
Assume k > 1, and view this group as a vector space.
There are 2k-1 vectors with 1 in the first component.
Since 2k-1 is even, this drops to zero,
and similarly for all other components.
The sum of the elements of G is 0.
If there is only one involution t,
the sum of the elements of G is t.