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Assume we have a finite group $G$, $|G|=p^cm$ where $p\not\mid m$ is prime. Fix a subgroup $H$ of order $p^a$ and a number $a\le b\le c$. Prove that the number of subgroups of order $p^b$ which contain the subgroup $H$ is $\equiv1\pmod p$.

I tried to use Wielandt's proof of Sylow's theorem as in Wielandt's proof of Sylow's theorem., but if we set $S=\{A\subset G, |A|=p^b,H\subset A\}$, we can't have $G$ act on $S$ by right multiplication. How can I prove this statement?

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  • $\begingroup$ I have not thought this through, but you could try replacing the set of group elements in Wielandt's proof by the set $C$ of $p^{c-a}m$ right cosets of $H$ in $G$, and the set $S$ by the set of subsets of $C$ of size $p^{b-a}$. $\endgroup$ – Derek Holt Nov 8 '18 at 8:06
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    $\begingroup$ See Theorem 7.9 in math.uconn.edu/~kconrad/blurbs/grouptheory/transitive.pdf. $\endgroup$ – KCd Nov 22 '18 at 11:28
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I suspect that this is something well-known, but I am not aware of any specific references. Anyway, here is a proof.

Theorem. Let $H$ be a $p$-subgroup of a group $G$, and suppose $p^t \geq |H|$. Then the number of subgroups $X \leq G$ such that $H \leq X$ and $|X|=p^t$ is congruent to $1 \bmod p$.

Lemma 1. The theorem is true in the case where $p^t$ is the order of a Sylow $p$-subgroup of $G$.

Proof. Let $H \leq P \in \operatorname{Syl}_p(G)$ and induct on $|P:H|$. The result is trivial if $H=P$, so we can assume that $H < P$ and we let $N = N_P(H)$. Conjugation by $N$ permutes the Sylow $p$-subgroups that contain $H$. If $S$ is one of these and $N \nleq S$, then $N$ does not normalise $S$, and so the $N$-orbit of $S$ has size divisible by $p$. Also, all members of this orbit fail to contain $N$, so we need only count the Sylow $p$-subgroups of $G$ that contain $N$, and since $N > H$, this number is congruent to $1 \bmod p$ by the inductive hypothesis.

Lemma 2. The theorem is true in the case where $G$ is a $p$-group.

Proof. Induct on $|G:H|$. The result is trivial if $H=G$ so we can assume that $H<G$. We can also assume that $p^t < |G|$. Let $\mathcal{X}$ and $\mathcal{M}$, respectively, be the set of subgroups of order $p^t$ that contain $H$ and the set of maximal subgroups that contain $H$. We count in two ways the number $N$ of ordered pairs $(X,M)$ with $X \in \mathcal{X}$, $M \in \mathcal{M}$ and $X \leq M$.

First, $N$ is the sum over $X \in \mathcal{X}$ of the number of maximal subgroups that contain $X$. Also, $N$ is the sum over $M \in \mathcal{M}$ of the number of members of $\mathcal{X}$ that are contained in $M$. It is easy to see that the number of maximal subgroups of $G$ that contain any given proper subgroup of $G$ is congruent to $1 \bmod p$. The first formula for $N$ gives $N \equiv |\mathcal{X}| \bmod p$ because the number of $M \in \mathcal{M}$ containing any given member of $\mathcal{X}$ is congruent to $1 \bmod p$. By the inductive hypothesis, the number of members of $\mathcal{X}$ contained in any given member of $\mathcal{M}$ is congruent to $1$, so the second formula for $N$ gives $$N \equiv |\mathcal{M}| \equiv 1 \bmod p.$$ Thus $$|\mathcal{X}| \equiv N \equiv 1 \bmod p,$$ as wanted.

Proof of Theorem. Let $\mathcal{X}$ be the set of subgroups of order $p^t$ that contain $H$ and let $\mathcal{P}$ be the set of $P \in \operatorname{Syl}_p(G)$ that contain $H$. We count the number $S$ of pairs $(X,P)$, where $X \in \mathcal{X}$, $P \in \mathcal{P}$ and $X \leq P$.

First, $S$ is the sum over $X \in \mathcal{X}$ of the number of $P \in \mathcal{P}$ such that $X \leq P$. Also, $S$ is the sum over $P \in \mathcal{P}$ of the number of $X \in \mathcal{X}$ such that $X \leq P$. Since the number of $P$ containing a given $X$ is congruent to $1 \bmod p$ by Lemma $\mathbf{1}$, the first formula for $S$ yields $S \equiv |\mathcal{X}| \bmod p$. The number of $X$ contained in a given $P$ is congruent to $1 \bmod p$ by Lemma $\mathbf{2}$, so the second formula for $S$ yields $S \equiv |\mathcal{P}| \bmod p$.

Also, $$|\mathcal{P}| \equiv 1 \bmod p$$ by Lemma $\mathbf{1}$, so we get $$|\mathcal{X}| \equiv S \equiv 1 \bmod p,$$ which concludes our proof.

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