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}