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In fact, if $|G|=p^n$ and $d(G)=d$, then $|\mathrm{Aut}(G)|$ divides the number $$(p^n-p^{n-d})(p^n-p^{n-d+1}) ... (p^n-p^{n-1}).$$ This result is due to P. Hall (1933)

Proof. The above product is the number of minimal bases of $G$. Howver, that number is a multiple of $|\text{Aut}(G)|$.

In fact, if $|G|=p^n$ and $d(G)=d$, then $|\mathrm{Aut}(G)|$ divides the number $$(p^n-p^{n-d})(p^n-p^{n-d+1}) ... (p^n-p^{n-1}).$$ This result is due to P. Hall (1933)

Proof. The above product is the number of minimal bases of $G$. However, that number is a multiple of $|\text{Aut}(G)|$. Indeed, if $\cal B$ is the set of all bases of $G$, then all $\text{Aut}(G)$-orbits on $\cal B$ have the same cardinality $|\text{Aut}(G)|$.

In fact, if $|G|=p^n$ and $d(G)=d$, then $|\mathrm{Aut}(G)|$ divides the number $$(p^n-p^{n-d})(p^n-p^{n-d+1}) ... (p^n-p^{n-1}).$$ This result is due to P. Hall (1933)

Y.B.

In fact, if $|G|=p^n$ and $d(G)=d$, then $|\mathrm{Aut}(G)|$ divides the number $$(p^n-p^{n-d})(p^n-p^{n-d+1}) ... (p^n-p^{n-1}).$$ This result is due to P. Hall (1933)

Proof. The above product is the number of minimal bases of $G$. Howver, that number is a multiple of $|\text{Aut}(G)|$.

In fact, if |G|=p^n and d(G)=d, then |Aut(G)| divide the number (p^n-p^{n-d})(p^n-p^{n-d+1) ... (p^n-p^{n-1}). This result is due to P.Hall (1933) Y.B.

In fact, if $|G|=p^n$ and $d(G)=d$, then $|\mathrm{Aut}(G)|$ divides the number $$(p^n-p^{n-d})(p^n-p^{n-d+1}) ... (p^n-p^{n-1}).$$ This result is due to P. Hall (1933)

Y.B.