Quaternion Group Presentation Assignment

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

Definition by presentation

The quaternion group has the following presentation:

The identity is denoted , the common element is denoted , and the elements are denoted respectively.

Confused about presentations in general or this one in particular? If you're new to this stuff, check out constructing quaternion group from its presentation. Sophisticated group theorists can read equivalence of presentations of dicyclic group

Verbal definitions

The quaternion group is a group with eight elements, which can be described in any of the following ways:

Multiplication table

In the table below, the row element is multiplied on the left and the column element on the right.

Position in classifications

Type of classification Name in that classification
GAP ID (8,4), i.e., the 4th among the groups of order 8
Hall-Senior number 5 among groups of order 8
Hall-Senior symbol

Elements

Conjugacy class structure

Automorphism class structure

Equivalence class (orbit) under action of automorphisms Size of equivalence class (orbit) Number of conjugacy classes in it Size of each conjugacy class Order of elements
1 1 1 1
1 1 1 2
6 3 2 4

Arithmetic functions

Basic arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 8#Arithmetic functions

Arithmetic functions of an element-counting nature

Arithmetic functions of a subgroup-counting nature

Lists of numerical invariants

Group properties

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 8#Group properties

Important properties

Other properties

Subgroups

Subgroup-defining functions and associated quotient-defining functions

Automorphisms and endomorphisms

Пока не везет.  - Она вздохнула.  - Быть может, придется ждать, пока Дэвид не найдет копию Танкадо.

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