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The modular group of order $p^{3}$ is extraspecial, of exponent $p^{2}$ when $p$ is odd

Statement

Let $p$ be a prime and let $M_{p} = A ⋊_{α} B$ be the modular group of order $p^{3}$ , with $A = ⟨ a ⟩$ of order $p^{2}$ , $B = ⟨ s ⟩$ of order $p$ and $s a s^{- 1} = a^{1 + p}$ . Then $∣ M_{p} ∣ = p^{3}$ , the group is nonabelian,

$Z (M_{p}) = [M_{p}, M_{p}] = ⟨ a^{p} ⟩$

has order $p$ , and $M_{p}$ is extraspecial of exponent $p^{2}$ . At $p = 2$ the group is the generalized dihedral group $Dih (C_{4})$ .

Facts & Assumptions

Given: A prime $p$ and the modular group $M_{p}$ with its generators $a$ and $s$ .

[F1]

The modular group of order $p^{3}$ is $M_{p} = A ⋊_{α} B$ with $A = ⟨ a ⟩$ of order $p^{2}$ , $B = ⟨ s ⟩$ of order $p$ , and $s a s^{- 1} = a^{1 + p}$ (The modular group of order $p^{3}$ as a semidirect product $C_{p^{2}} ⋊ C_{p}$ ).

[F2]

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

[F3]

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

[F4]

For a finite group $G$ , $\exp (G) = \min {n \in N : n > 0 and g^{n} = e for every g \in G}$ (The exponent of a finite group).

[L1]

In $N ⋊_{α} H$ the sets $\overset{ˉ}{N} = {(n, 1)}$ and $\overset{ˉ}{H} = {(1, h)}$ are subgroups, $\overset{ˉ}{N}$ is normal, $\overset{ˉ}{N} \cap \overset{ˉ}{H} = {(1, 1)}$ , every element has a unique factorisation $(n, 1) (1, h)$ , and $(1, h) (n, 1) (1, h)^{- 1} = (α_{h} (n), 1)$ (The canonical copy of $N$ is normal, the canonical copy of $H$ is a complement, and conjugation induces the action).

[L2]

The class of $1 + p$ in $Z / p^{2}$ is a unit of multiplicative order $p$ , and $(1 + p)^{k} = 1 + k p$ (Raising to the power $1 + p$ is an automorphism of order $p$ of a cyclic group of order $p^{2}$ ).

[L3]

For a finite $p$ -group $P$ the following are equivalent: $P$ is extraspecial; $P$ is nonabelian, $∣ Z (P) ∣ = p$ and $P / Z (P)$ is elementary abelian; $P$ is nonabelian and $Z (P) = P^{'} = Φ (P)$ has order $p$ (Three equivalent descriptions of an extraspecial $p$ -group).

[L4]

For a finite group $G$ and $H \leq G$ , $∣ G ∣ = [G : H] ∣ H ∣$ (Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

[L5]

If the quotient group $G / Z (G)$ is cyclic, then $G$ is abelian (If $G / Z (G)$ is cyclic, then $G$ is abelian).

[L6]

For $N ⊴ G$ , the quotient $G / N$ is abelian if and only if $[G, G] \subseteq N$ ( $G / N$ is abelian if and only if $[G, G] \subseteq N$ ).

[L7]

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

[L8]

For $n \geq 1$ , $Dih (C_{n}) = C_{n} ⋊ C_{2}$ where the nonidentity element of $C_{2}$ acts by inversion ( $Dih (C_{n}) = C_{n} ⋊ C_{2}$ with inversion action has order $2 n$ and the dihedral relations).

[L9]

$⟨ S ⟩$ is the smallest subgroup of $G$ containing $S$ (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

[L10]

For all $m, n \in Z$ one has $(g^{m})^{n} = g^{m n}$ (Exponent laws in a group: $g^{m + n} = g^{m} g^{n}$ and $(g^{m})^{n} = g^{m n}$ for all $m, n \in Z$ , and $(g h)^{n} = g^{n} h^{n}$ when $g$ and $h$ commute).

Proof

technique · direct

1.1

Every element of $M_{p}$ has a unique factorisation $a^{i} s^{j}$ with $0 \leq i < p^{2}$ and $0 \leq j < p$ , so $∣ M_{p} ∣ = p^{2} \cdot p = p^{3}$ and $M_{p}$ is a finite $p$ -group generated by $a$ and $s$ .

F1L1L9

1.2

The relation gives $[s, a] = s a s^{- 1} a^{- 1} = a^{1 + p} a^{- 1} = a^{p}$ , and $a^{p}$ has order $p$ because $a$ has order $p^{2}$ .

F1F3L2L10

1.3

The element $a^{p}$ is central: it commutes with $a$ , and $s a^{p} s^{- 1} = (s a s^{- 1})^{p} = a^{(1 + p) p} = a^{p + p^{2}} = a^{p}$ .

F1F2L1L2L10

2.1

$M_{p}$ is nonabelian, since $s a s^{- 1} = a^{1 + p} \neq a$ : equality would give $a^{p} = 1$ , contradicting that $a$ has order $p^{2}$ .

F1L2step 1.1

2.2

Modulo $⟨ a^{p} ⟩$ the images of $a$ and $s$ commute, by step 1.2, and they generate; so $M_{p} / ⟨ a^{p} ⟩$ is abelian and $[M_{p}, M_{p}] \leq ⟨ a^{p} ⟩$ . With $a^{p} = [s, a] \in [M_{p}, M_{p}]$ this gives $[M_{p}, M_{p}] = ⟨ a^{p} ⟩$ , of order $p$ .

F3L6L9step 1.1step 1.2step 1.3

3.1

The centre contains $⟨ a^{p} ⟩$ and is not all of $M_{p}$ ; its order divides $p^{3}$ , and an order of $p^{2}$ would leave a quotient of order $p$ , necessarily cyclic, forcing $M_{p}$ abelian. So $Z (M_{p}) = ⟨ a^{p} ⟩$ has order $p$ and equals $[M_{p}, M_{p}]$ .

F2L4L5step 1.3step 2.1step 2.2

3.2

The exponent is $p^{2}$ : the element $a$ has order $p^{2}$ , so the exponent is a multiple of $p^{2}$ dividing $p^{3}$ ; it is not $p^{3}$ , because an element of order $p^{3}$ in a group of order $p^{3}$ would generate it and make it cyclic, hence abelian.

F4L4L9step 1.1step 2.1

4.1

The quotient $M_{p} / Z (M_{p})$ has order $p^{2}$ , is abelian because $[M_{p}, M_{p}] = Z (M_{p})$ , and is not cyclic, so all of its nonidentity elements have order $p$ and it is elementary abelian; by the second description in the characterisation $M_{p}$ is extraspecial.

L3L4L5L6L7step 2.1step 3.1

5.1

At $p = 2$ the relation reads $s a s^{- 1} = a^{3} = a^{- 1}$ , so the action of $B$ on $A$ is inversion and $M_{2}$ is by definition the semidirect product of a cyclic group of order four by a cyclic group of order two acting by inversion, which is $Dih (C_{4})$ .

F1L8L2step 1.1∎

Remarks

The group is extraspecial at every prime and its exponent is $p^{2}$ at every prime; what changes at $p = 2$ is only that the resulting group already has a name, since inversion is the unique nontrivial power automorphism of a cyclic group of order four.

Depends on

Used by

Dependency tree · two levels

71 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. van Beek, Topics in Finite p-Groups, Proposition 2.32(i) and 2.32(iv) (standard reference, not scraped)
D. A. Craven, The Theory of p-Groups, Definition 3.3 (standard reference, not scraped)

The modular group of order p3p3 is extraspecial, of exponent p2p2 when pp is odd