Modular maximal-cyclic group

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Definition

Let $n$ be a natural number greater than or equal to $4$. The modular maximal-cyclic group or modular group of order $2^{n}$, denoted $M_{2^{n}}$ or $M_{n}(2)$, is defined by the following presentation:

$\!M_{2^{n}}=M_{n}(2):=\langle a,x\mid a^{2^{n-1}}=x^{2}=e,xax=a^{2^{n-2}+1}\rangle$

(here, $e$ is the symbol for the identity element).

Particular cases

$n$	$2^{n}$	$n-1$	$2^{n-1}$	Group
4	16	3	8	modular maximal-cyclic group:M16
5	32	4	16	modular maximal-cyclic group:M32
6	64	5	32	modular maximal-cyclic group:M64
7	128	6	64	modular maximal-cyclic group:M128

See also

• Semidihedral group, a group family with a very similar definition.