There exists an abelian subgroup of maximum order whose normalizer contains every abelian subgroup it normalizes
Statement
Suppose is a prime number and is a group of prime power order with underlying prime . Denote by the set of abelian subgroups of maximum order. Then, there exists such that its normalizer in , namely , contains every abelian subgroup of normalized by . In particular, contains every abelian normal subgroup of and hence contains the join of all abelian normal subgroups of .
Related facts
Similar facts
- Thompson's replacement theorem for abelian subgroups
- Any abelian normal subgroup normalizes an abelian subgroup of maximum order: This is a weaker version of the statement that is easier to prove, and follows directly from Thompson's replacement theorem.
Stronger facts in some cases
Note that this fact is uninteresting for small orders for the following silly reason: for small orders, it is also true that, among the abelian subgroups of maximum order, there exists a normal subgroup. The existence of a normal subgroup that is abelian of maximum order is obviously substantially stronger than this statement.
Below, we indicate which of a bunch of stronger statements is true, and why. In particular, we note from this that the smallest examples of interest where the stronger statements do not hold are for the prime 2 and (or possibly higher for some , depending on other replacement results that are dependent on the value of -- see Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one for instance) for odd primes.
Value so we're considering groups of order | Is every abelian subgroup of maximum order normal? | Does there always exist an abelian subgroup of maximum order which is normal? | Explanation |
---|---|---|---|
1 | Yes | Yes | All groups of order are abelian, so the whole group is the unique abelian subgroup of maximum order. |
2 | Yes | Yes | All groups of order are abelian, so the whole group is the unique abelian subgroup of maximum order. |
3 | Yes | Yes | If the group is abelian, it is itself the unique subgroup of maximum order, which is normal. Otherwise, there exist subgroups of order , all of which are abelian and normal. |
4 | Yes | Yes | If the group is abelian, it is itself the unique subgroup of maximum order, which is normal. Otherwise, by the existence of abelian normal subgroups of small prime power order, there exist abelian subgroups of order and index . Using the fact that subgroup of index equal to least prime divisor of group order is normal (or instead that nilpotent implies every maximal subgroup is normal) we conclude that all these subgroups are normal. |
5 | No (?) | Yes | If the maximum order for an abelian subgroup is or , then every subgroup of that order is normal. If the maximum order is or smaller, we can use the existence of abelian normal subgroups of small prime power order to conclude that there exists an abelian normal subgroup of the same order. |
6 | No (?) | Yes | If the maximum order for an abelian subgroup is or , we can use that every subgroup of that order is normal. If the maximum order is , use abelian-to-normal replacement theorem for prime-square index. If the maximum order is or less, use existence of abelian normal subgroups of small prime power order. |
7 | No | Yes | If the maximum order for an abelian subgroup is or , we can use that every subgroup of that order is normal. If the maximum order is , use abelian-to-normal replacement theorem for prime-square index. If the maximum order is or less (actually, it cannot be less), use existence of abelian normal subgroups of small prime power order. |
8 | No | Yes | Order is taken care of because prime power order implies every maximal subgroup is normal. For an odd prime , the abelian-to-normal replacement theorem for prime-cube index for odd prime as well as abelian-to-normal replacement for prime-square index show that if there exists an abelian subgroup of order or , there is an abelian normal subgroup of the same order. Further, the existence of abelian normal subgroups of small prime power order takes care of order up to . Note: The statement is also true for but the in case needs to be proved more indirectly and cannot directly use the replacement theorem for prime-cube index. |
9 | No | No for Yes for odd primes |
Order is taken care of because prime power order implies every maximal subgroup is normal. For an odd prime , the abelian-to-normal replacement theorem for prime-cube index for odd prime as well as [[abelian-to-normal replacement for prime-square index show that if there exists an abelian subgroup of order or , there is an abelian normal subgroup of the same order. Finally, the Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime handles orders up to . For the prime , see Abelian-to-normal replacement fails for prime-cube index for prime equal to two. The smallest example for that is a subgroup of order in a group of order . |