Groups of order 21: Difference between revisions
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{{groups of order|21}} | {{groups of order|21}} | ||
==Statistics at a glance== | |||
{| class="sortable" border="1" | |||
! Quantity !! Value !! Explanation | |||
|- | |||
| Total number of groups || [[count::2]] || | |||
|- | |||
| Number of abelian groups || [[abelian count::1]] || | |||
|} | |||
==The groups== | |||
There are, up to isomorphism, two groups of order 21, indicated in the table below: | There are, up to isomorphism, two groups of order 21, indicated in the table below: | ||
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! Group !! GAP ID (second part) !! Abelian? | ! Group !! GAP ID (second part) !! Abelian? | ||
|- | |- | ||
| [[ | | [[General semilinear group:GammaL(1,8)]] || 1 || No | ||
|- | |- | ||
| [[cyclic group:Z21]] || 2 || Yes | | [[cyclic group:Z21]] || 2 || Yes | ||
|} | |} | ||
This is classified by the [[classification of groups of order a product of two distinct primes]] in the case with two isomorphism classes since <math>21 = 3 \cdot 7</math>, with <math>3 \mid 7-1</math>. | |||
Z7⋊Z3 is the smallest non-abelian group of odd order. | Z7⋊Z3 is the smallest non-abelian group of odd order. | ||
Latest revision as of 00:55, 18 November 2023
This article gives information about, and links to more details on, groups of order 21
See pages on algebraic structures of order 21 | See pages on groups of a particular order
Statistics at a glance
| Quantity | Value | Explanation |
|---|---|---|
| Total number of groups | 2 | |
| Number of abelian groups | 1 |
The groups
There are, up to isomorphism, two groups of order 21, indicated in the table below:
| Group | GAP ID (second part) | Abelian? |
|---|---|---|
| General semilinear group:GammaL(1,8) | 1 | No |
| cyclic group:Z21 | 2 | Yes |
This is classified by the classification of groups of order a product of two distinct primes in the case with two isomorphism classes since , with .
Z7⋊Z3 is the smallest non-abelian group of odd order.