Order is product of Mersenne prime and one more implies normal Sylow subgroup
Contents
Statement
Suppose is a prime number such that
is a Mersenne prime (?). Consider:
.
Then, any group of order has a nontrivial Normal Sylow subgroup (?): either the
-Sylow subgroup is normal, or the
-Sylow subgroup is normal. Thus,
admits a Sylow tower (?), so
is a Group having a Sylow tower (?).
Examples
Some examples for small Mersenne primes are as follows:
-
: In a group of order
, either the
-Sylow subgroup or the
-Sylow subgroup is normal.
-
, In a group of order
, either the
-Sylow subgroup or the
-Sylow subgroup is normal.
-
: In a group of order
, either the
-Sylow subgroup or the
-Sylow subgroup is normal.
Related facts
Cannot pinpoint to any one prime
We can find groups of order where only the
-Sylow subgroup is normal, and we can find groups where only the other subgroup is normal. Here are examples of both:
- Only the
-Sylow subgroup is normal: Consider the general affine group
. This is the semidirect product of the additive group of a field of order
with its multiplicative group. Here, only the
-Sylow subgroup is normal.
- Only the
-Sylow subgroup is normal: Consider the direct product of the dihedral group of order
and a cyclic group of order
. Here, only the
-Sylow subgroup is normal.
Facts used
Proof
Given: A Mersenne prime , a group
of order
.
To prove: Either the -Sylow subgroup or the
-Sylow subgroup of
is normal.
Proof: Let and let
and
denote the number of
-Sylow subgroups and
-Sylow subgroups respectively. We have:
.
Also, we have, since equals the index of the normalizer of a
-Sylow subgroup:
This forces either or
. If
, we are done. Consider the case
. Let
be all the
-Sylow subgroups. Since these are all groups of prime order, any two of them intersect trivially. Thus, the total number of non-identity elements in the subgroups is:
.
Thus, there are exactly elements that are not among the non-identity elements of
-Sylow subgroups. Note that any element in a
-Sylow subgroup cannot be among these non-identity elements of
-Sylow subgroups, so any element in a
-Sylow subgroup must be among these
elements. This forces any
-Sylow subgroup to be contained in this set of
elements. Since the order of a
-Sylow subgroup is
, this forces there to be a unique
-Sylow subgroup -- precisely those
elements. This completes the proof.
References
Textbook references
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, Page 147, Section 4.5 (Sylow's theorem), Exercise 13, More info