Extraspecial $p$-Groups and Central Products — Examples

Given: The objects and hypotheses in the Example.

For an extraspecial $p$-group $P$ with $Z(P)=⟨z⟩$, the commutator pairing is the map $b_z(\bar{x},\bar{y})\in \mathbb{Z}\mathrm{/}p$ determined by $[x,y]=z^{b_z(\bar{x},\bar{y})}$ (The commutator pairing of an extraspecial $p$-group relative to a chosen generator of its centre).

The commutator pairing is independent of the coset representatives, is ${\mathbb{F}}_p$-bilinear on $P\mathrm{/}Z(P)$, and is alternating (The commutator pairing is well defined on the central quotient, is bilinear over ${\mathbb{F}}_p$, and is alternating).

The generalized dihedral group $\mathrm{Dih}(C_4)$ and the quaternion group $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively ($\mathrm{Dih}(C_4)$ and $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively).

$\mathrm{Dih}(A)=A⋊C_2\mathrm{.}$ ( The generalized dihedral group $\mathrm{Dih}(A)=A⋊C_2$ for an abelian group $A$).

$Q_8:=\{1,-1,i,-i,j,-j,k,-k\}\subseteq {\mathbb{H}}^{\times }\mathrm{.}$ (The quaternion group $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ inside the nonzero quaternions).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

Given: The objects and hypotheses in the Example.

The Heisenberg group of order $p^3$ is the set $(\mathbb{Z}\mathrm{/}p{)}^3$ with $(a,b,c)(a^{\mathrm{\prime }},b^{\mathrm{\prime }},c^{\mathrm{\prime }})=(a+a^{\mathrm{\prime }},b+b^{\mathrm{\prime }},c+c^{\mathrm{\prime }}+a{b}^{\mathrm{\prime }})$ (The Heisenberg group of order $p^3$ over $\mathbb{Z}\mathrm{/}p$).

The Heisenberg multiplication makes $(\mathbb{Z}\mathrm{/}p{)}^3$ a nonabelian group of order $p^3$ (The Heisenberg multiplication is a group law, nonabelian, on a set of $p^3$ elements).

The Heisenberg group of order $p^3$ is extraspecial, and for odd $p$ it has exponent $p$ (The Heisenberg group of order $p^3$ is extraspecial, and for odd $p$ it has exponent $p$).

For a finite group $G$, its exponent is $\exp (G)=\min \{n\in \mathbb{N}:n>0\text{ and }{g}^n=e\text{ for every }g\in G\}\mathrm{.}$ The set is nonempty by, and gives its least member; powers use. (The exponent of a finite group).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

$\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }\mathrm{.}$ (Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$).

Given: The objects and hypotheses in the Example.

The Heisenberg group of order $p^3$ is the set $(\mathbb{Z}\mathrm{/}p{)}^3$ with $(a,b,c)(a^{\mathrm{\prime }},b^{\mathrm{\prime }},c^{\mathrm{\prime }})=(a+a^{\mathrm{\prime }},b+b^{\mathrm{\prime }},c+c^{\mathrm{\prime }}+a{b}^{\mathrm{\prime }})$ (The Heisenberg group of order $p^3$ over $\mathbb{Z}\mathrm{/}p$).

The Heisenberg multiplication makes $(\mathbb{Z}\mathrm{/}p{)}^3$ a nonabelian group of order $p^3$ (The Heisenberg multiplication is a group law, nonabelian, on a set of $p^3$ elements).

The Heisenberg group of order $p^3$ is extraspecial, and for odd $p$ it has exponent $p$ (The Heisenberg group of order $p^3$ is extraspecial, and for odd $p$ it has exponent $p$).

Every nonabelian group of order $p^3$ is extraspecial (A nonabelian group of order $p^3$ is extraspecial).

The generalized dihedral group $\mathrm{Dih}(C_4)$ and the quaternion group $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively ($\mathrm{Dih}(C_4)$ and $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively).

The two nonabelian groups of order eight are $\mathrm{Dih}(C_4)$ and $Q_8$ (For each prime there are exactly two nonabelian groups of order $p^3$ up to isomorphism).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

Given: The objects and hypotheses in the Example.

The modular group of order $p^3$ is the external semidirect product $C_{p^2}{⋊}_{\alpha }{C}_p$ for the order-$p$ automorphism $a\mapsto a^{1+p}$ (The modular group of order $p^3$ as a semidirect product $C_{p^2}⋊C_p$).

For every prime $p$ the map $a\mapsto a^{1+p}$ is an automorphism of order $p$ of a cyclic group of order $p^2$ (Raising to the power $1+p$ is an automorphism of order $p$ of a cyclic group of order $p^2$).

The modular group of order $p^3$ is extraspecial, and its exponent is $p^2$ (The modular group of order $p^3$ is extraspecial, of exponent $p^2$ when $p$ is odd).

For a finite group $G$, its exponent is $\exp (G)=\min \{n\in \mathbb{N}:n>0\text{ and }{g}^n=e\text{ for every }g\in G\}\mathrm{.}$ The set is nonempty by, and gives its least member; powers use. (The exponent of a finite group).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

$\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }\mathrm{.}$ (Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$).

Given: The objects and hypotheses in the Example.

The generalized dihedral group $\mathrm{Dih}(C_4)$ and the quaternion group $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively ($\mathrm{Dih}(C_4)$ and $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively).

If $P_1$ and $P_2$ are extraspecial $2$-groups with $t_i$ solutions of $x^2=1$, then $P_1\circ P_2$ has $(t_1{t}_2+(\mathrm{\mid }{P}_1\mathrm{\mid }-t_1)(\mathrm{\mid }{P}_2\mathrm{\mid }-t_2))\mathrm{/}2$ such solutions (A product formula for the number of square roots of the identity in a central product of extraspecial $2$-groups).

$Q_8\circ Q_8\cong \mathrm{Dih}(C_4)\circ \mathrm{Dih}(C_4)$ ($Q_8\circ Q_8\cong \mathrm{Dih}(C_4)\circ \mathrm{Dih}(C_4)$).

For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$ up to isomorphism, with $2^{2n}+2^n$ and $2^{2n}-2^n$ solutions of $x^2=1$ (For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

A set $A$ is finite when $A\approx n$ for some $n\in \mathbb{N}$. (The cardinality $\mid A\mid$ of a finite set).

A central product of extraspecial $2$-groups identified along their centres is extraspecial and has order $\mathrm{\mid }{E}_1\mathrm{\mid }\mathrm{\mid }{E}_2\mathrm{\mid }\mathrm{/}2$ (A central product of extraspecial $p$-groups identified along their centres is extraspecial).

Given: The central product $\mathrm{Dih}(C_4)\circ \mathrm{Dih}(C_4)$ and its two canonical factor maps.

For $n\ge 1$, $\mathrm{Dih}(C_n)=C_n⋊C_2$ has order $2n$, and with $C_n=⟨r⟩$ and $C_2=⟨s⟩$ every element has a unique form $r^i$ or $r^{i}s$ with $0\le i<n$ ( $\mathrm{Dih}(C_n)=C_n⋊C_2$ with inversion action has order $2n$ and the dihedral relations).

The canonical maps into a central product are injective homomorphisms whose images commute elementwise, generate the whole central product, and meet in the common central line (The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre).

$⟨S⟩:=\bigcap \{H:H\le G\text{ and }S\subseteq H\}\mathrm{.}$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups).

$\mathrm{Dih}(C_4)$ is extraspecial of order eight, and a central product of two extraspecial $2$-groups along their centres is extraspecial of order $\mathrm{\mid }{E}_1\mathrm{\mid }\mathrm{\mid }{E}_2\mathrm{\mid }\mathrm{/}2$ ($\mathrm{Dih}(C_4)$ and $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively, A central product of extraspecial $p$-groups identified along their centres is extraspecial).

Given: The objects and hypotheses in the Example.

For groups $G,H$ with central subgroups $Z_1\le Z(G)$, $Z_2\le Z(H)$ and an isomorphism $\alpha :Z_1\to Z_2$, the central product $G{\circ }_{\alpha }H$ is the quotient of $G\times H$ by $N=\{(z,\alpha (z{)}^{-1}):z\in Z_1\}$ (The central product $G{\circ }_{\alpha }H$ of two groups along an isomorphism of central subgroups).

The subgroup $N=\{(z,\alpha (z{)}^{-1}):z\in Z_1\}$ of $G\times H$ is central, hence normal (The identified subgroup used to form a central product is central, hence normal).

The canonical maps $G\to G{\circ }_{\alpha }H$ and $H\to G{\circ }_{\alpha }H$ are injective homomorphisms whose images commute elementwise, generate $G{\circ }_{\alpha }H$, and meet in the image of $Z_1$ (The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre).

$\mathrm{\mid }G{\circ }_{\alpha }H\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{\mid }H\mathrm{\mid }\mathrm{/}\mathrm{\mid }{Z}_1\mathrm{\mid }$, the centre of $G{\circ }_{\alpha }H$ is the image of $Z(G)\times Z(H)$, and its derived subgroup is the image of $G^{\mathrm{\prime }}\times H^{\mathrm{\prime }}$ (Order, centre and derived subgroup of a central product).

If $G=⟨g⟩$ is cyclic, then exactly one of the following applies: (Every cyclic group is isomorphic to $(\mathbb{Z},+)$ or to $(\mathbb{Z}\mathrm{/}n,+)$ for its finite order $n\ge 1$).

$G\times H:=\{(g,h):g\in G,h\in H\}$ (The external direct product $G\times H$ with componentwise multiplication).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

Given: The objects and hypotheses in the Example.

Every maximal abelian subgroup of an extraspecial $p$-group of order $p^{1+2n}$ has order $p^{1+n}$ (In an extraspecial $p$-group of order $p^{1+2n}$ every maximal abelian subgroup has order $p^{1+n}$).

The generalized dihedral group $\mathrm{Dih}(C_4)$ and the quaternion group $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively ($\mathrm{Dih}(C_4)$ and $Q_8$ are extraspecial of order $8$, with six and two solutions of $x^2=1$ respectively).

For $n\ge 1$, $\mathrm{Dih}(C_n)=C_n⋊C_2$ has order $2n$, and with $C_n=⟨r⟩$ and $C_2=⟨s⟩$ one has $r^n=s^2=1$, $sr{s}^{-1}=r^{-1}$, and every element has a unique form $r^i$ or $r^{i}s$ with $0\le i<n$ ( $\mathrm{Dih}(C_n)=C_n⋊C_2$ with inversion action has order $2n$ and the dihedral relations).

$\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }\mathrm{.}$ (Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$).

Given: The proposed claim together with the witness named in the Statement refuted.

A finite $p$-group $P$ is special when $Z(P)=P^{\mathrm{\prime }}=\mathrm{\Phi }(P)$ is elementary abelian, and extraspecial when in addition $P$ is nonabelian and this common subgroup has order $p$ (Special and extraspecial $p$-groups).

The Heisenberg group of order $p^3$ is the set $(\mathbb{Z}\mathrm{/}p{)}^3$ with $(a,b,c)(a^{\mathrm{\prime }},b^{\mathrm{\prime }},c^{\mathrm{\prime }})=(a+a^{\mathrm{\prime }},b+b^{\mathrm{\prime }},c+c^{\mathrm{\prime }}+a{b}^{\mathrm{\prime }})$ (The Heisenberg group of order $p^3$ over $\mathbb{Z}\mathrm{/}p$).

The Heisenberg multiplication makes $(\mathbb{Z}\mathrm{/}p{)}^3$ a nonabelian group of order $p^3$ (The Heisenberg multiplication is a group law, nonabelian, on a set of $p^3$ elements).

The Heisenberg group of order $p^3$ is extraspecial, and for odd $p$ it has exponent $p$ (The Heisenberg group of order $p^3$ is extraspecial, and for odd $p$ it has exponent $p$).

$G\times H:=\{(g,h):g\in G,h\in H\}$ (The external direct product $G\times H$ with componentwise multiplication).

An elementary abelian $p$-group is a finite abelian $p$-group in which every nonidentity element has order $p$; the trivial group is permitted (,, ). (Elementary abelian $p$-groups).

$Z(G):=\{z\in G:zg=gz\text{ for every }g\in G\}\mathrm{.}$ (The center $Z(G)$ of a group).

$[g,h]:=gh{g}^{-1}{h}^{-1}\mathrm{.}$ (Commutators $[g,h]=gh{g}^{-1}{h}^{-1}$ and the commutator subgroup $[G,G]$).

For every finite $p$-group $P$, the Frattini formula gives $\mathrm{\Phi }(P)=P^{\mathrm{\prime }}{P}^p\mathrm{.}$ ($\mathrm{\Phi }(P)=P^{\mathrm{\prime }}{P}^p$ for a finite $p$-group)

For a group $G$ and a prime $p$, the $p$th-power subgroup is $G^p=⟨g^p:g\in G⟩\mathrm{.}$ (The $p$th-power subgroup $G^p$)

For finite groups $G$ and $H$, the direct product has order $\mathrm{\mid }G\times H\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{\mid }H\mathrm{\mid }$. (For finite groups $G$ and $H$, $\mathrm{\mid }G\times H\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{\mid }H\mathrm{\mid }$)

Given: The proposed claim together with the witness named in the Statement refuted.

Subgroups $G_1,\dots ,G_r$ of $G$ form an internal central product when they generate $G$ and $[G_i,G_j]=1$ for $i\mathrm{\ne }j$ (Internal central products of a finite family of subgroups).

Subgroups $G_1,\dots ,G_r$ form an internal central product of $G$ if and only if the multiplication map $G_1\times \cdots \times G_r\to G$ is a surjective homomorphism each of whose factors meets its kernel trivially (Internal central products are the images of external ones).

$Q_8\circ Q_8\cong \mathrm{Dih}(C_4)\circ \mathrm{Dih}(C_4)$ ($Q_8\circ Q_8\cong \mathrm{Dih}(C_4)\circ \mathrm{Dih}(C_4)$).

For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$ up to isomorphism, with $2^{2n}+2^n$ and $2^{2n}-2^n$ solutions of $x^2=1$ (For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$).

$Q_8:=\{1,-1,i,-i,j,-j,k,-k\}\subseteq {\mathbb{H}}^{\times }\mathrm{.}$ (The quaternion group $Q_8=\{\pm 1,\pm i,\pm j,\pm k\}$ inside the nonzero quaternions).

$\mathrm{Dih}(A)=A⋊C_2\mathrm{.}$ ( The generalized dihedral group $\mathrm{Dih}(A)=A⋊C_2$ for an abelian group $A$).

Given: The proposed claim together with the witness named in the Statement refuted.

A finite $p$-group $P$ is special when $Z(P)=P^{\mathrm{\prime }}=\mathrm{\Phi }(P)$ is elementary abelian, and extraspecial when in addition $P$ is nonabelian and this common subgroup has order $p$ (Special and extraspecial $p$-groups).

$Z(G):=\{z\in G:zg=gz\text{ for every }g\in G\}\mathrm{.}$ (The center $Z(G)$ of a group).

Given: The proposed claim together with the witness named in the Statement refuted.

For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$ up to isomorphism, with $2^{2n}+2^n$ and $2^{2n}-2^n$ solutions of $x^2=1$ (For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$).

For odd $p$ and each $n\ge 1$ there are exactly two extraspecial groups of order $p^{1+2n}$ up to isomorphism, one of exponent $p$ and one of exponent $p^2$ (For odd $p$ and each $n\ge 1$ there are exactly two extraspecial groups of order $p^{1+2n}$, distinguished by their exponent).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).

For a finite group $G$, its exponent is $\exp (G)=\min \{n\in \mathbb{N}:n>0\text{ and }{g}^n=e\text{ for every }g\in G\}\mathrm{.}$ The set is nonempty by, and gives its least member; powers use. (The exponent of a finite group).

Given: The proposed claim together with the witness named in the Statement refuted.

For an extraspecial $p$-group $P$ with $Z(P)=⟨z⟩$, the commutator pairing is the map $b_z(\bar{x},\bar{y})\in \mathbb{Z}\mathrm{/}p$ determined by $[x,y]=z^{b_z(\bar{x},\bar{y})}$ (The commutator pairing of an extraspecial $p$-group relative to a chosen generator of its centre).

The commutator pairings of $\mathrm{Dih}(C_4)$ and $Q_8$ agree, while the groups are not isomorphic (The commutator pairings of $\mathrm{Dih}(C_4)$ and $Q_8$ are the same, while the groups are not isomorphic).

Given: The proposed claim together with the witness named in the Statement refuted.

For an extraspecial $p$-group $P$ with $Z(P)=⟨z⟩$, the commutator pairing is the map $b_z(\bar{x},\bar{y})\in \mathbb{Z}\mathrm{/}p$ determined by $[x,y]=z^{b_z(\bar{x},\bar{y})}$ (The commutator pairing of an extraspecial $p$-group relative to a chosen generator of its centre).

The commutator pairing is independent of the coset representatives, is ${\mathbb{F}}_p$-bilinear on $P\mathrm{/}Z(P)$, and is alternating (The commutator pairing is well defined on the central quotient, is bilinear over ${\mathbb{F}}_p$, and is alternating).

$Z(G):=\{z\in G:zg=gz\text{ for every }g\in G\}\mathrm{.}$ (The center $Z(G)$ of a group).

For every prime $p$, the operations of addition and multiplication on $\mathbb{Z}\mathrm{/}p$ make it a field. (For every prime $p$, the two operations on $\mathbb{Z}\mathrm{/}p$ make it a field).

Given: The proposed claim together with the witness named in the Statement refuted.

The centre of an extraspecial $p$-group has no complement (The centre of an extraspecial $p$-group has no complement).

In this situation $H$ is called a complement to $N$ in $G$. (An internal semidirect product and a complement to a normal subgroup).

Given: The proposed claim together with the witness named in the Statement refuted.

Every extraspecial $p$-group is an internal central product of nonabelian subgroups of order $p^3$ pairwise intersecting in its centre (Every extraspecial $p$-group is an internal central product of nonabelian subgroups of order $p^3$).

An extraspecial $p$-group has order $p^{1+2n}$ for some $n\ge 1$ (An extraspecial $p$-group has order $p^{1+2n}$ for some $n\ge 1$).

The order of a finite group. Let $G$ be a group whose underlying set is finite, so that $G\approx n$ for some $n\in \mathbb{N}$. (The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity).