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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.

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    $\begingroup$ 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. $\endgroup$
    – Geoff Robinson
    Commented Nov 18, 2016 at 15:40
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    $\begingroup$ 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. $\endgroup$
    – Geoff Robinson
    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. $\endgroup$
    – maryam
    Commented Nov 18, 2016 at 17:55
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    $\begingroup$ 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. $\endgroup$
    – Geoff Robinson
    Commented Nov 18, 2016 at 18:02
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    $\begingroup$ A structure theorem for finite nonsolvable groups with a maximal Sylow $2$-subgroup was proved by Bernd Baumann, J. Alg. 38 #1 (1976). $\endgroup$
    – Richard Lyons
    Commented Mar 11, 2017 at 16:39

2 Answers 2

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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;
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    $\begingroup$ Thanks. I think this number is rather amazing, since it is possible to exclude all p-groups of order less than 768. $\endgroup$
    – maryam
    Commented Nov 18, 2016 at 18:35
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    $\begingroup$ 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. $\endgroup$
    – zibadawa timmy
    Commented Nov 18, 2016 at 18:53
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    $\begingroup$ 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 $O_{2}(G) = 1.$ $\endgroup$
    – Geoff Robinson
    Commented Oct 18 at 12:06
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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.

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  • $\begingroup$ 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. $\endgroup$
    – Geoff Robinson
    Commented Oct 18 at 12:10

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