The Gao constant 
of a finite group 
is the least integer 
such that every sequence of 
elements of 
contains 
terms that can be reordered to have product equal to the
identity element. Repetitions are allowed, and
the selected terms need not be consecutive (Qu 2026). The required subsequence length
is the group order, not the group
exponent.
For a cyclic group of order 
,
the Erdős-Ginzburg-Ziv theorem
gives 
. Qu (2026) proved that every finite noncyclic
group satisfies
 |
where 
is the least prime divisor of 
. Equality holds iff 
has a subgroup
that is cyclic and has subgroup
index 
. If no such subgroup exists,
the stronger bound 
holds.
Qu (2026) reports AI assistance with proofs involving finite simple groups.
The Erdős-Ginzburg-Ziv constant instead requires a zero-sum subsequence whose length is the group exponent. These two prescribed lengths agree for a cyclic group.
