4 research outputs found
On the number of epi-, mono-, and homomorphisms of groups
It is known that the number of homomorphisms from a group to a group
is divisible by the greatest common divisor of the order of and the
exponent of . We investigate the number of homomorphisms satisfying
some natural conditions such as injectivity or surjectivity. The simplest
nontrivial corollary of our results is the following fact: {\it in any finite
group, the number of generating pairs such that , is a
multiple of the greatest common divisor of 15 and the order of the group
.Comment: 5 pages. A Russian version of this paper is at
http://halgebra.math.msu.su/staff/klyachko/papers.htm . V2: minor
corrections. arXiv admin note: text overlap with arXiv:1806.0887
On the number of tuples of group elements satisfying a first-order formula
Our result contains as special cases the Frobenius theorem (1895) on the
number of solutions to the equation in a finite group and the Solomon
theorem (1969) on the number of solutions in a group to systems of equations
with fewer equations than unknowns. We consider arbitrary first-order formulae
in the group language with constants instead of systems of equations. Our
result generalizes substantially a theorem of Klyachko and Mkrtchyan (2014) on
this topic