R-a-jones at 10:50, 22 October 2023

2023-10-22T10:50:25Z

← Older revision		Revision as of 10:50, 22 October 2023
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	{{groups of order\|65}}		{{groups of order\|65}}

	Up to isomorphism, there is a ''unique'' group of order 15, namely [[cyclic group:Z65]], which is also the [[external direct product]] of [[cyclic group:Z5]] and [[cyclic group:Z13]].		Up to isomorphism, there is a ''unique'' group of order 65, namely [[cyclic group:Z65]], which is also the [[external direct product]] of [[cyclic group:Z5]] and [[cyclic group:Z13]].

	The fact of uniqueness follows from the [[classification of groups of order a product of two distinct primes]]. Since <math>15 = 5 \cdot 13</math> and <math>5</math> does not divide <math>(13 - 1)</math>, the number <math>65</math> falls in the ''one isomorphism class'' case.		The fact of uniqueness follows from the [[classification of groups of order a product of two distinct primes]]. Since <math>15 = 5 \cdot 13</math> and <math>5</math> does not divide <math>(13 - 1)</math>, the number <math>65</math> falls in the ''one isomorphism class'' case.

	Another way of viewing this is that <math>65</math> is a [[cyclicity-forcing number]], i.e., any group of order 65 is cyclic. See the [[classification of cyclicity-forcing numbers]] to see the necessary and sufficient condition for a natural number to be cyclicity-forcing.		Another way of viewing this is that <math>65</math> is a [[cyclicity-forcing number]], i.e., any group of order 65 is cyclic. See the [[classification of cyclicity-forcing numbers]] to see the necessary and sufficient condition for a natural number to be cyclicity-forcing.

R-a-jones: Created page with "{{groups of order|65}} Up to isomorphism, there is a ''unique'' group of order 15, namely cyclic group:Z65, which is also the external direct product of cyclic grou..."

2023-10-22T10:48:36Z

Created page with "{{groups of order|65}} Up to isomorphism, there is a ''unique'' group of order 15, namely cyclic group:Z65, which is also the external direct product of cyclic grou..."

New page

{{groups of order|65}}

Up to isomorphism, there is a ''unique'' group of order 15, namely [[cyclic group:Z65]], which is also the [[external direct product]] of [[cyclic group:Z5]] and [[cyclic group:Z13]].

The fact of uniqueness follows from the [[classification of groups of order a product of two distinct primes]]. Since <math>15 = 5 \cdot 13</math> and <math>5</math> does not divide <math>(13 - 1)</math>, the number <math>65</math> falls in the ''one isomorphism class'' case.

Another way of viewing this is that <math>65</math> is a [[cyclicity-forcing number]], i.e., any group of order 65 is cyclic. See the [[classification of cyclicity-forcing numbers]] to see the necessary and sufficient condition for a natural number to be cyclicity-forcing.

Groups of order 65 - Revision history

R-a-jones at 10:50, 22 October 2023

R-a-jones: Created page with "{{groups of order|65}} Up to isomorphism, there is a ''unique'' group of order 15, namely cyclic group:Z65, which is also the external direct product of cyclic grou..."