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

Facts used

Equivalence of definitions of finite nilpotent group: Specifically, the fact that in a finite nilpotent group, all the Sylow subgroups are normal
Normality is upper join-closed: If $N \leq H, L \leq G$ and $N$ is normal in both $H$ and $L$ , then $N$ is normal in $G$
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 $π = {p_{1}, p_{2}, \dots, p_{r}}$ </math> 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 $π$ .

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 $π$ 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	$H = P_{1} P_{2} \dots P_{r}, K = Q_{1} Q_{2} \dots Q_{r}$ 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 $π$ -Hall in $G$	Step (1)
3	Consider the subgroups $P_{1} P_{2} \dots P_{r - 1}$ and $Q_{1} Q_{2} \dots Q_{r - 1}$ . These are both nilpotent $π ∖ {p_{r}}$ -Hall subgroups of $G$ .			Steps (1), (2)	Step-direct
4	$P_{1} P_{2} \dots P_{r - 1}$ and $Q_{1} Q_{2} \dots Q_{r - 1}$ 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 $L = P_{1} P_{2} \dots P_{r - 1} J$ where $J$ is a $p_{r}$ -Sylow subgroup of $G$ .		$K$ is nilpotent	Steps (1), (4)
6	Suppose $T = ⟨ H, L ⟩$ 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 $⟨ H, L ⟩ = G$ . Then, $N : = P_{1} P_{2} \dots P_{r - 1}$ 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 $⟨ H, L ⟩ = G$ .
8	Suppose $⟨ H, L ⟩ = G$ . Construct $N$ as in Step (7). Let $\bar{G} = G / N, \bar{H} = H / N, \bar{L} = L / N$ . Then, $\bar{H}$ and $\bar{L}$ are $p_{r}$ -Sylow subgroups of $\bar{G}$ .			Steps (5), (7)	direct from the construction
9	Suppose $⟨ H, L ⟩ = G$ . Continuing notation from Step (8), $\bar{H}$ and $\bar{L}$ are conjugate in $\bar{G}$ by the image $\bar{g}$ of some element $g \in G$ in $\bar{G} = G / N$ . 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 $⟨ H, L ⟩ \neq G$ . Step (9) shows that $H$ and $L$ are conjugate if $⟨ H, L ⟩ = G$ . 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.

References

Textbook references

Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 236, Exercise 2 (Chapter 6)