If all elements of a group
are contained in the set-theoretic union of proper subgroups
, then we define this
collection to be a cover of
.
When such a cover exists, the cardinality of the smallest possible cover is called the covering
number of
, denoted
by
. Maróti
determined
for odd
and provided an estimate
for even
. The second
author later determined
for
when
, while joint work
of the second author with Kappe and Nikolova-Popova also verified that Maróti’s rule holds for
and established the
covering numbers of
for various other small
.
Currently,
is the
smallest value for which
is unknown. In this paper, we prove the covering number of
is
.
Keywords
symmetric groups, finite union of proper subgroups,
subgroup covering
