# Nilpotent Hall subgroups of same order are conjugate

## History

The result was proved by Wielandt in 1954.

## Statement

### Verbal statement

In a finite group, any two nilpotent Hall subgroups of the same order are conjugate subgroups.

### Statement with symbols

Suppose $G$ is a finite group and $\pi$ is a set of primes. Let $H$ and $K$ be Hall $\pi$-subgroups of $G$. Then, $H$ and $K$ are conjugate subgroups inside $G$: in other words, there exists $g \in G$ such that $gHg^{-1} = K$.

## Facts used

1. Equivalence of definitions of finite nilpotent group: Specifically, the fact that in a finite nilpotent group, all the Sylow subgroups are normal
2. Normality is upper join-closed: If $N \le H, L \le G$ and $N$ is normal in both $H$ and $L$, then $N$ is normal in $G$
3. Sylow implies order-conjugate: Specifically, the fact that any two $p$-Sylow subgroups of a finite group are conjugate

## Proof

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Given: A finite group $G$, a set $\pi = \{ p_1,p_2 , \dots, p_r\}$[/itex] of prime divisors of $G$. Two nilpotent Hall subgroups $H,K$ of $G$

To prove: $H$ and $K$ are conjugate.

Proof: We prove the result by a double induction: induction on the order of $G$, and, for a given $G$, induction on the size of $\pi$.

Outer induction on order: Let's assume the result is true for all groups of order smaller than the order of $G$.

Inner induction on size of set of primes: Note that the result is true if $\pi$ has size one, by Fact (3). The hard part is the inductive step. We will assume that the result holds for the particular group $G$ for all smaller sized sets of primes.

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 $H = P_1P_2 \dots P_r, K = Q_1Q_2 \dots Q_r$ where $P_i,Q_i$ are $p$-Sylow subgroups of $H$ and $K$ respectively. Further, $P_i, Q_i$ are normal in $H,K$ respectively. Fact (1) $H,K$ are nilpotent. Follows directly from Fact (1).
2 $P_i,Q_i$ are both $p_i$-Sylow subgroups of $G$. $H,K$ are $\pi$-Hall in $G$ Step (1)
3 Consider the subgroups $P_1P_2\dots P_{r-1}$ and $Q_1Q_2 \dots Q_{r-1}$. These are both nilpotent $\pi \setminus \{ p_r \}$-Hall subgroups of $G$. Steps (1), (2) Step-direct
4 $P_1P_2 \dots P_{r-1}$ and $Q_1Q_2 \dots Q_{r-1}$ are conjugate in $G$. inductive assumption on the size of the set of primes Steps (2), (3) direct from the step and the inductive assumption.
5 We can use the conjugating element of Step (4) to conjugate $K$ to a nilpotent subgroup $L$ of $G$ that looks like $L = P_1P_2 \dots P_{r-1}J$ where $J$ is a $p_r$-Sylow subgroup of $G$. $K$ is nilpotent Steps (1), (4)
6 Suppose $T = \langle H, L \rangle$ is a proper subgroup of $G$. Then, $H$ and $L$ are conjugate in $T$, hence also in $G$. inductive assumption on the order of the group, $H$ is nilpotent Step (5) Direct from the inductive assumption
7 Suppose $\langle H, L \rangle = G$. Then, $N := P_1P_2 \dots P_{r-1}$ is normal in $G$. Fact (2) $N$ is normal in $H$ and in $L$ on account of being a Hall subgroup of a nilpotent group. Hence, it is normal in $\langle H, L \rangle = G$.
8 Suppose $\langle H, L \rangle = G$. Construct $N$ as in Step (7). Let $\overline{G} = G/N, \overline{H} = H/N, \overline{L} = L/N$. Then, $\overline{H}$ and $\overline{L}$ are $p_r$-Sylow subgroups of $\overline{G}$. Steps (5), (7) direct from the construction
9 Suppose $\langle H,L \rangle = G$. Continuing notation from Step (8), $\overline{H}$ and $\overline{L}$ are conjugate in $\overline{G}$ by the image $\overline{g}$ of some element $g \in G$ in $\overline{G} = G/N$. Then $g$ must conjugate $H$ to $L$. Fact (3) Steps (7), (8) direct from Step (8).
10 $H$ and $L$ are conjugate. Steps (6), (9) Step (6) shows that $H$ and $L$ are conjugate if $\langle H, L \rangle \ne G$. Step (9) shows that $H$ and $L$ are conjugate if $\langle H,L \rangle = G$. Thus, $H$ and $L$ are conjugate in every case.
11 $H$ and $K$ are conjugate. Steps (5), (10) Step (5) says that $K$ and $L$ are conjugate. Step (10) says that $H$ and $L$ are conjugate. Since conjugacy is an equivalence relation, we obtain that $H$ and $L$ are conjugate.