Groups of order 65: Difference between revisions
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{{groups of order|65}} | {{groups of order|65}} | ||
Up to isomorphism, there is a ''unique'' group of order | 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. | ||
Latest revision as of 10:50, 22 October 2023
This article gives information about, and links to more details on, groups of order 65
See pages on algebraic structures of order 65 | See pages on groups of a particular order
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 and does not divide , the number falls in the one isomorphism class case.
Another way of viewing this is that 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.