About subgroups of the order of the normalizer of a p-subgroup

group-theorysylow-theory

I recently came across a doubt I've already partially exposed in this post without getting a solution. So I'd like to isolate my final concern from the textbook I mentioned there.
The real doubt is:

Is true that in a group $G$, if $P$ is a Sylow $p$-subgroup and $N$ is the normalizer of $P$, then the only subgroups of order $|N|$ are the normalizers of the Sylow p-subgroups?

I've checked within $A_5$ and that seems true in that case:

there $6$ 5-sylow, with their related $6$ normalizers of order $10$, which are exhausting all subgroups of order $10$ in $A_5$
there are $10$ 3-sylow, with their related $10$ normalizers of order 6, which are exhausting all subgroups of order $6$ in $A_5$
there are $5$ 2-sylow of order $4$, which their related $5$ normalizers of order $12$, which are exhausting all subgroups of order $12$ in $A_5$

My doubt is this holds true for any generic group $G$ or just for $A_5$.

Thanks in advance.

Best Answer

No, a counterexample is $\mathtt{SmallGroup}(324,160)$, which has the structure $3^3\!:\!A_4$. It has a unique minimal normal subgroup $K$ of order $3^3$ and $G/K \cong A_4$.

The normalizer $N$ of a Sylow $2$-subgroup has order $12$ with $N \cong A_4$, but there is another conjugacy class of subgroups of order $12$, which do not have a normal Sylow $2$-subgroup.

Alternating

The Sylow 2-subgroups of the symmetric groups are direct products of wreath products of Sylow 2-subgroups of $S_2$ -- this was known in the 19th century. The Sylow 2-subgroups of the alternating groups are index 2 subgroups.

For $n=4m+2$ and $n=4m+3$, the copies of $S_{4m}$ inside $A_n$ have odd index $2m+1$ or $(4m+3)(2m+1)$, so the Sylow 2-subgroup of $S_{4m}$ is isomorphic to the Sylow 2-subgroups of $A_{4m+2}$ and $A_{4m+3}$.

Weisner (1925) computes the order of the normalizers of the Sylow $p$-subgroups of symmetric and alternating groups (so counts them). The main result for us is that Sylow 2-subgroups are self-normalizing in simple alternating groups (except $A_5$ with normalizer $A_4$).

Weir (1955) computes the characteristic subgroups of the Sylow $p$-subgroup of the symmetric groups, but only for odd $p$. Lewis (1968) modifies this to handle $p=2$ for both symmetric and alternating groups. Dmitruk–Suščanskʹkiĭ (1981) take the approach of Kaloujnine (1945-1948), again handling $p=2$ and alternating groups.

Harada–Lang (2005) observes that the Sylow 2-subgroups of $A_{4m}$ and $A_{4m+1}$ are directly indecomposable (while those of $A_{4m+2}$ and $A_{4m+3}$ are directly indecomposable iff $m$ is a power of $2$).

Weisner, Louis; “On the Sylow Subgroups of the Symmetric and Alternating Groups.” Amer. J. Math. 47 (1925), no. 2, 121–124. MR1506549 DOI:10.2307/2370639
Kaloujnine, Léo “La structure des p-groupes de Sylow des groupes symétriques finis.” Ann. Sci. École Norm. Sup. (3) 65, (1948). 239–276. ^{_{Also see: C. R. Acad. Sci. Paris
221 (1945), 222–224; ibid.
222 (1946), 1424–1425; ibid.
223 (1946), 703–705; ibid.
224 (1947), 253–255.}}
Weir, A. J. “The Sylow subgroups of the symmetric groups.” Proc. Amer. Math. Soc. 6 (1955), 534–541. MR72142 DOI:10.2307/2033425
Lewis, Robert Edward. “On the Sylow two-subgroups of the alternating groups.” Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. 1968. 48 pp. MR2617989
Dmitruk, Ju. V.; Suščanskʹkiĭ, V. Ī. “Construction of Sylow 2-subgroups of alternating groups and normalizers of Sylow subgroups in symmetric and alternating groups.” Ukrain. Mat. Zh. 33 (1981), no. 3, 304–312. MR621637
Harada, Koichiro; Lang, Mong Lung. “Indecomposable Sylow 2-subgroups of simple groups.” Acta Appl. Math. 85 (2005), no. 1-3, 161–194. MR2128910 DOI10.1007/s10440-004-5618-0

Classical groups in odd characteristic

There is a huge difference in Sylow $p$-subgroup structure depending on whether $p$ is the characteristic of the field. In this section we assume $p$ is not the characteristic of the field.

In case $p$ is not the characteristic, then Weir (1955) showed that symmetric groups and classical groups are very similar, but again $p=2$ was left out until Carter-Fong (1964), and then more uniformly in Wong (1967). Algorithms to handle all Sylow $p$-subgroups of classical groups are described in Stather (2008).

The gist is that in GL, GO, GU, and Sp, the Sylow $p$-subgroups are direct products of wreath products of cyclic groups of order $p$ with the Sylow $p$-subgroup of the two-dimensional groups. For SL, SO or $\Omega$, SU the answers are more complicated, but only because an easy to understand part has been chopped off the top.

Weir, A. J. “Sylow $p$-subgroups of the classical groups over finite fields with characteristic prime to $p$.” Proc. Amer. Math. Soc. 6 (1955), 529–533. MR72143 DOI:10.2307/2033424
Carter, Roger; Fong, Paul. “The Sylow 2-subgroups of the finite classical groups.” J. Algebra 1 (1964) 139–151. MR166271 DOI:10.1016/0021-8693(64)90030-4
Wong, Warren J. “Twisted wreath products and Sylow 2−subgroups of classical simple groups.” Math. Z. 97 (1967) 406–424. MR212095 DOI:10.1007/BF01112816
Stather, Mark. “Constructive Sylow theorems for the classical groups.” J. Algebra 316 (2007), no. 2, 536–559. MR2356843 DOI:10.1016/j.jalgebra.2007.03.013

Exceptional groups in odd characteristic

Sylow 2-subgroups for finite groups of Lie type are similar to the classical case: there is a 2-dimensional group $P_2$ and a “top” group $X$ (which need not be $C_2 \wr C_2 \wr \ldots \wr C_2$, but that is probably the correct picture to have) such that the $X$-conjugates of $P_2$ are commute with each other, so that $X \ltimes P_2^n$ is a Sylow 2-subgroup. The $P_2$ are the Sylow 2-subgroups of the so-called “fundamental subgroups” of Aschbacher (1977), where we view groups of Lie type as built up from rank 1 groups, in this case commuting rank 1 subgroups isomorphic to SL2. These are used in Aschbacher (1980) to describe groups in which a Sylow 2-subgroup is contained in a unique maximal subgroup, and Harada–Lang (2005) describes which Sylow 2-subgroups are indecomposable. GLS I.A.4.10 covers Aschbacher's ideas as well.

Aschbacher, Michael. “A characterization of Chevalley groups over fields of odd order.” Ann. of Math. (2) 106 (1977), no. 2, 353–398. MR498828
Aschbacher, Michael. “A characterization of Chevalley groups over fields of odd order. II.” Ann. of Math. (2) 106 (1977), no. 3, 399–468. MR498829
Aschbacher, Michael. “On finite groups of Lie type and odd characteristic.” J. Algebra 66 (1980), no. 2, 400–424. MR593602 DOI:10.1016/0021-8693(80)90095-2
Gorenstein, Daniel; Lyons, Richard; Solomon, Ronald. The classification of the finite simple groups. Number 3. Part I. Chapter A. Almost simple K-groups. Mathematical Surveys and Monographs, 40.3. American Mathematical Society, Providence, RI, 1998. xvi+419 pp. ISBN: 0-8218-0391-3 MR1490581
Harada, Koichiro; Lang, Mong Lung. “Indecomposable Sylow 2-subgroups of simple groups.” Acta Appl. Math. 85 (2005), no. 1-3, 161–194. MR2128910 DOI10.1007/s10440-004-5618-0

Groups in characteristic 2

Here the Sylow 2-subgroups are basically groups of upper triangular matrices and are often best understood in terms of linear algebra. Weir (1955) describes the characteristic subgroups and those normalized by important subgroups of GL. These general ideas work in all the groups of Lie type. The main description I know is Chevalley's commutator formula, as explained in Carter (1972).

XXX: Decent reference for the classical, and then the exceptional. Maybe specifically handle Suzuki and Ree.

Weir, A. J. “Sylow $p$-subgroups of the general linear group over finite fields of characteristic $p$.” Proc. Amer. Math. Soc. 6, (1955). 454–464. MR69175 DOI:10.2307/2032788
Carter, Roger W. Simple groups of Lie type. Pure and Applied Mathematics, Vol. 28. John Wiley & Sons, London-New York-Sydney, 1972. viii+331 pp. Mr407163

Sporadic

I think each one is a special snowflake. XXX: Lookup coincidences in Sylow structure.

Proof that in a simple group of order 168 the intersection of the normalizers of two distinct 7-sylows is of order 3

Your argument would be fine, if your goal were to show that $N_G(P)\cap N_G(P')$ must have order $3$ for at least some pair of Sylow $7$-subgroups $P,P'$. However, the way I read the task is that you are to prove that $|N_G(P)\cap N_G(P')|=3$ for any pair of distinct Sylow $7$-subgroups.

More about that later. Revisiting the argument and setting up the scene for the stronger claim.

A possibly simpler way to get to the point you reached yourself could go as follows. Let $X$ be the set of Sylow $7$-subgroups. You correctly deduced that $|X|=8$ and that hence $|N_G(P)|=21$ for all $P\in X$. Therefore, by Cauchy, there is an element $z$ of order three in $N_G(P)$. Consider the conjugation action of $H=\langle z\rangle$ on $X$.

The orbits of $H$ on $X$ have sizes $1$ or $3$.
Because $|X|=8\equiv2\pmod3$, there must be at least two orbits of size $1$.
Clearly $\{P\}$ is an orbit of size one. If $\{Q\}$ is another, both $P$ and $Q$ are normalized by $H$. In particular $H\le N_G(P)\cap N_G(Q)$.
Because $N_G(P)$ and $N_G(Q)$ cannot share elements of order $7$, their intersection cannot have order $>3$, so $H=N_G(P)\cap N_G(Q)$.

We can then proceed and show that $|N_G(P)\cap N_G(P')|=3$ for all $P,P'\in X$, $P\neq P'$.

We saw above (you showed this in a different way) that there is another Sylow $7$, $Q\in X$ such that $N_G(P)\cap N_G(Q)$ has order three.
Let $x$ be a generator of $P$. We know that $x$ does not normalize any Sylow $7$-subgroups other than $P$.
Clearly $$x(N_G(P)\cap N_G(Q))x^{-1}=N_G(xPx^{-1})\cap N_G(xQx^{-1})=N_G(P)\cap N_G(xQx^{-1}),$$ so we see that $N_G(xQx^{-1})$ also intersects $N_G(P)$ in a subgroup of order three (that must be $xHx^{-1}$).
Repeating the above with powers of $x$ we see that the same holds for every $P'\in X$ that belongs to the $P$-orbit of $Q$.
But the $P$-orbit of $Q$ in $X$ must have seven elements. Hence it contains all the Sylow $7$-subgroups other than $P$ itself.

At this point we have proven that $N_G(P)$ intersects the normalizers of all the other Sylow $7$-groups in a subgroup of order three. Because we started with an arbitrary $P\in X$, the claim holds for all pairs of Sylow $7$s.

The last step would also follow from the fact that the conjugation action of $G$ on $X$ is transitive.

May be many, if not all, of the steps I wanted to add were obvious to you. I just think that in a first course on this theme you would be expected to include them. Nothing deep going on there.