A finite elementary Abelian group is a finite Abelian group in which every nonidentity element has
the same prime order 
.
It is isomorphic to 
for an integer 
, and its addition is that of a vector
space over the finite field with 
elements. The trivial group
is often included by allowing 
. For example, the Klein four-group is elementary Abelian
for 
.
The order complex in the Quillen
conjecture uses the nontrivial elementary Abelian subgroups
of a given finite group, ordered by inclusion.
Elementary Abelian Group
See also
Abelian Group, Finite Field, Finite Group, Klein Four-Group, Order Complex, Quillen ConjectureExplore with Wolfram|Alpha
References
Milne, J. S. Group Theory, ver. 4.01. 2025. https://www.jmilne.org/math/CourseNotes/GT.pdf.Piterman, K. I. and Smith, S. D. "Some Results on Quillen's Conjecture via Equivalent-Poset Techniques." J. Comb. Algebra 9, 265-387, 2025. https://doi.org/10.4171/JCA/95.Cite this as:
Weisstein, Eric W. "Elementary Abelian Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ElementaryAbelianGroup.html