Cyclic group of prime-square order is not an algebra group for odd prime
This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.
View other such facts for p-groups|View other such facts for finite groups
- Algebra group is isomorphic to algebra subgroup of unitriangular matrix group of degree one more than logarithm of order to base of field size
By Fact (1), if is an algebra group over , it must be isomorphic to a subgroup of . However, has exponent if is odd, so , which has exponent , cannot be isomorphic to a subgroup of it.
- MathOverflow question: p-groups realisable as 1+J,where J is a nilpotent finite F-Algebra: Jack Schmidt's answer mentions this fact, and sketches a proof for the similar fact that Z8 is not an algebra group