Using combinatorial methods, we will examine squares of conjugacy classes in the symmetric groups
S
ν
{S_\nu }
of all permutations of an infinite set of cardinality
ℵ
ν
{\aleph _\nu }
. For arbitrary permutations
p
∈
S
ν
p \in {S_\nu }
, we will characterize when each element
s
∈
S
ν
s \in {S_\nu }
with finite support can be written as a product of two conjugates of
p
p
, and if
p
p
has infinitely many fixed points, we determine when all elements of
S
ν
{S_\nu }
are products of two conjugates of
p
p
. Classical group-theoretical theorems are obtained from similar results.