In: Advanced Math
A) Suppose a group G has order 35 and acts on a set S consisting of four elements. What can you say about the action?
B) What happens if |G|=28? |G|=30?
A) This action induces a group homomorphism
of
into the group of permutations
of
,
given by
for all
and
.
Note that
, so that
. If
then by first isomorphism theorem, we know that
By Lagrange's theorem, we get
, so that
divides
. Again,
since
is a subgroup, by Lagrange,
divides
. Therefore,
divides
, so
that
divides
.
Thus,
,
which implies
. Thus, the action
is trivial, given by
for all
and
.
B)
Suppose that . The same
argument as above shows that
divides
, so
that
divides
.
Thus, there are three possibilities:
If
holds then the action is trivial.
Suppose that . The same
argument as above shows that
divides
, so
that
divides
.
Thus, there are three possibilities:
If
holds then the action is trivial.