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Marius Tarnauceanu
Marius Tarnauceanu
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Let $G$ be an extraspecial $2$-group, i.e. $Z(G)=G'=\Phi(G)$ has order $2$. Then $|G|=2^{2n+1}$ for some $n\geq 1$, $G\cong D_8^{*n}$ or $G\cong Q_8*D_8^{*n}$$G\cong Q_8*D_8^{*(n-1)}$, and $G$ has one of the following presentations: $$\langle x_1,y_1, ..., x_n,y_n\mid [x_i,x_j]=[y_i,y_j]=1, [x_i,y_j]=1 \mbox{ for } i\neq j,$$ $$\hspace{-10mm}[x_i,y_i]=z \mbox{ and } x_i^2=y_i^2=z^2=1 \mbox{ for } z\in Z(G)\rangle$$or
$$\langle x_1,y_1, ..., x_n,y_n\mid [x_i,x_j]=[y_i,y_j]=1, [x_i,y_j]=1 \mbox{ for } i\neq j,$$ $$\hspace{-6mm}[x_i,y_i]=z \mbox{ and } y_i^2=z, x_i^2=z^2=1 \mbox{ for } z\in Z(G)\rangle.$$Also, $G$ possesses elementary abelian subgroups of order at most $2^{n+1}$. Is it known an explicit formula for the number of elementary abelian subgroups of order $2^k$, $k=1,...,n+1$, of $G$ containing $Z(G)$?

Let $G$ be an extraspecial $2$-group, i.e. $Z(G)=G'=\Phi(G)$ has order $2$. Then $|G|=2^{2n+1}$ for some $n\geq 1$, $G\cong D_8^{*n}$ or $G\cong Q_8*D_8^{*n}$, and $G$ has one of the following presentations: $$\langle x_1,y_1, ..., x_n,y_n\mid [x_i,x_j]=[y_i,y_j]=1, [x_i,y_j]=1 \mbox{ for } i\neq j,$$ $$\hspace{-10mm}[x_i,y_i]=z \mbox{ and } x_i^2=y_i^2=z^2=1 \mbox{ for } z\in Z(G)\rangle$$or
$$\langle x_1,y_1, ..., x_n,y_n\mid [x_i,x_j]=[y_i,y_j]=1, [x_i,y_j]=1 \mbox{ for } i\neq j,$$ $$\hspace{-6mm}[x_i,y_i]=z \mbox{ and } y_i^2=z, x_i^2=z^2=1 \mbox{ for } z\in Z(G)\rangle.$$Also, $G$ possesses elementary abelian subgroups of order at most $2^{n+1}$. Is it known an explicit formula for the number of elementary abelian subgroups of order $2^k$, $k=1,...,n+1$, of $G$ containing $Z(G)$?

Let $G$ be an extraspecial $2$-group, i.e. $Z(G)=G'=\Phi(G)$ has order $2$. Then $|G|=2^{2n+1}$ for some $n\geq 1$, $G\cong D_8^{*n}$ or $G\cong Q_8*D_8^{*(n-1)}$, and $G$ has one of the following presentations: $$\langle x_1,y_1, ..., x_n,y_n\mid [x_i,x_j]=[y_i,y_j]=1, [x_i,y_j]=1 \mbox{ for } i\neq j,$$ $$\hspace{-10mm}[x_i,y_i]=z \mbox{ and } x_i^2=y_i^2=z^2=1 \mbox{ for } z\in Z(G)\rangle$$or
$$\langle x_1,y_1, ..., x_n,y_n\mid [x_i,x_j]=[y_i,y_j]=1, [x_i,y_j]=1 \mbox{ for } i\neq j,$$ $$\hspace{-6mm}[x_i,y_i]=z \mbox{ and } y_i^2=z, x_i^2=z^2=1 \mbox{ for } z\in Z(G)\rangle.$$Also, $G$ possesses elementary abelian subgroups of order at most $2^{n+1}$. Is it known an explicit formula for the number of elementary abelian subgroups of order $2^k$, $k=1,...,n+1$, of $G$ containing $Z(G)$?

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Marius Tarnauceanu
Marius Tarnauceanu
  • 71
  • 2

Number of elementary abelian subgroups of extraspecial $2$-groups

Let $G$ be an extraspecial $2$-group, i.e. $Z(G)=G'=\Phi(G)$ has order $2$. Then $|G|=2^{2n+1}$ for some $n\geq 1$, $G\cong D_8^{*n}$ or $G\cong Q_8*D_8^{*n}$, and $G$ has one of the following presentations: $$\langle x_1,y_1, ..., x_n,y_n\mid [x_i,x_j]=[y_i,y_j]=1, [x_i,y_j]=1 \mbox{ for } i\neq j,$$ $$\hspace{-10mm}[x_i,y_i]=z \mbox{ and } x_i^2=y_i^2=z^2=1 \mbox{ for } z\in Z(G)\rangle$$or
$$\langle x_1,y_1, ..., x_n,y_n\mid [x_i,x_j]=[y_i,y_j]=1, [x_i,y_j]=1 \mbox{ for } i\neq j,$$ $$\hspace{-6mm}[x_i,y_i]=z \mbox{ and } y_i^2=z, x_i^2=z^2=1 \mbox{ for } z\in Z(G)\rangle.$$Also, $G$ possesses elementary abelian subgroups of order at most $2^{n+1}$. Is it known an explicit formula for the number of elementary abelian subgroups of order $2^k$, $k=1,...,n+1$, of $G$ containing $Z(G)$?