3
$\begingroup$

Thanks for any help or comments.

Suppose that $G$ is a finite group. A Carter subgroup of $G$ is a nilpotent self-normalizing subgroup of $G$. Carter and Vdovin have shown that solvable groups have Carter subgroups, and that in addition, in every group with Carter subgroups, the Carter subgroups are conjugate -- see

Carter, R. W. (1961), Nilpotent selfnormalizing subgroups of soluble groups, Mathematische Zeitschrift, 75 (2): 136–139.

Vdovin, E. P. (2006), On the conjugacy problem for Carter subgroups. (Russian.), Sibirsk. Mat. Zh., 47 (4): 725–730. Translation in Siberian Math. J. 47 (2006), no. 4, 597–600

Vdovin, E. P. (2007), Carter subgroups in finite almost simple groups. (Russian.), Algebra i Logika, 46 (2): 157–216.

My question is about the structure of groups whose Carter subgroups are their Sylow $2$-subgroups? I mean, I am interested in any theorem which guides me towards some classification of this type of groups.

Improve this question
$\endgroup$
5
  • 1
    $\begingroup$ You are asking to classify finite groups with a self-normalizing Sylow $2$-subgroup. The so-called $C(G,T)$-Theorem of M. Aschbacher may be relevant here. $\endgroup$ Commented Nov 18, 2016 at 15:40
  • 1
    $\begingroup$ To expand: a first step might be to consider finite groups in which Sylow $2$-subgroup is maximal, where the Theorem of Aschbacher mentioned above shoudl certainly be helpful. $\endgroup$ Commented Nov 18, 2016 at 15:43
  • $\begingroup$ @Geoff, thaks. I am not familar with Aschbacher theorem. At least do you think is it possible to characterize groups are minimal with respect to this property? I mean groups contain self-normalizing 2-sylow subgroup but every subgroup does not contain such subgroup. $\endgroup$ Commented Nov 18, 2016 at 17:55
  • 1
    $\begingroup$ I think that that these minimal groups in particular have a maximal Sylow $2$-subgroup, and I do think that groups with a maximal Sylow $2$-subgroups may be attackable as I suggested above. $\endgroup$ Commented Nov 18, 2016 at 18:02
  • 2
    $\begingroup$ A structure theorem for finite nonsolvable groups with a maximal Sylow $2$-subgroup was proved by Bernd Baumann, J. Alg. 38 #1 (1976). $\endgroup$ Commented Mar 11, 2017 at 16:39

2 Answers 2

Reset to default
1
$\begingroup$

Classifying finite groups with self-normalizing Sylow $2$-subgroup seems rather hopeless as $10608361$ of the $10625619$ groups of order less than $768$ have this property. --

A GAP function to count the groups of order $n$ with this property is as follows:

NrOfGroupsWithSelfNormalizingSylow2Subgroup := function ( n ) if n = 1 then return 1; elif SmallestRootInt(n) = 2 then return NrSmallGroups(n); elif n mod 2 = 1 then return 0; else return Number(AllGroups(n), G -> Size(SylowSubgroup(G,2)) = Size(Normalizer(G,SylowSubgroup(G,2)))); fi; end;
Improve this answer
$\endgroup$
3
  • 1
    $\begingroup$ Thanks. I think this number is rather amazing, since it is possible to exclude all p-groups of order less than 768. $\endgroup$ Commented Nov 18, 2016 at 18:35
  • 1
    $\begingroup$ Almost all of those groups are the groups of order 512, though, which is a potentially misleading result. I don't think it makes sense to apply the question at hand to p-groups. It's trivial in those cases. $\endgroup$ Commented Nov 18, 2016 at 18:53
  • 1
    $\begingroup$ There is no doubt that it is pretty hopeless to classify $2$-groups of a given $2$-power order, and while the question allows these, it may be that the intended spirit of the question is less concerned about $2$-groups. A finite group $G$ has a self-normalizing Sylow $2$-subgroup if and only if $G/O_{2}(G)$ has that property. So it might be more instructive to consider finite groups $G$ with a self-normalizing Sylow $2$-subgroup and with $O_{2}(G) = 1.$ $\endgroup$ Commented Oct 18, 2024 at 12:06
4
$\begingroup$

Here's some stronger evidence in favor of Stefan's claim that such groups are too numerous and varied to expect a classification.

glist := AllSmallGroups(Size,[2..511],IsPGroup,false,G->Size(G) mod 2 = 0, true, G->SylowSubgroup(G,2)=Normalizer(G,SylowSubgroup(G,2)));; Length(glist); Sum(List(Filtered([2..511],n->(n mod 2 = 0) and (not IsPrimePowerInt(n))),n->NrSmallGroups(n)));

This shows that of the 33510 non-p-groups with even order less than 512, 25673 of them have self-normalizing Sylow 2-subgroups. Moreover, given $n\leq 7$, every 2-group of order $2^n$ appears as the Sylow 2-subgroup in some such example.

syls:=Set(List(glist,G->IdGroup(SylowSubgroup(G,2))));; ForAll([1..7],n->NrSmallGroups(2^n) = Number(syls,P->P[1]=2^n));

So not only does it look like we can expect a large proportion of groups to have this property, but that we cannot even reasonably expect to constrain the structure of the Sylow 2-subgroup.

Improve this answer
$\endgroup$
1
  • $\begingroup$ You may get a different type of statistic if you look at groups $G$ with $O_{2}(G) = 1$ and with a self-normalizing Sylow $2$-subgroup. $\endgroup$ Commented Oct 18, 2024 at 12:10

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.