JP Journal of Algebra, Number Theory and Applications
Volume 3, Issue 1, Pages 13 - 26
(April 2003)
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NONABELIAN GROUPS WITH PERFECT ORDER SUBSETS
Carrie E. Finch (U. S. A.) and Lenny Jones (U. S. A.)
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Abstract: Let G be a finite
group and let x
Î
G. Define the order subset of G
determined by x to be the set of all
elements in G that have the same order as
x. A group G is said to have
perfect order subsets if the number of
elements in each order subset of G is a
divisor of |G|.
In
this article we prove a theorem for a class of
nonabelian groups, which is analogous to Theorem
4 in [Amer. Math. Monthly 109 (2002), 509-516].
We then prove that there are infinitely many
nonabelian groups with perfect order subsets. In
addition, all values of q are determined
such that the special linear group, SL(2,
q), has
perfect order subsets. Next, we give a
discussion of some necessary conditions for
general nonabelian groups to have perfect order
subsets. We conclude by stating some
conjectures. |
Keywords and phrases: group, conjugacy class, exponential
Diophantine equations. |
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