Groups of order 45

This article gives information about, and links to more details on, groups of order 45
See pages on algebraic structures of order 45 | See pages on groups of a particular order

Up to isomorphism, there are two groups of order $45$. Both are abelian.

Another way of viewing this is that $45$ is a abelian-forcing number, i.e., any group of order $45$ is abelian. See the classification of abelian-forcing numbers to see the necessary and sufficient condition for a natural number to be abelian-forcing.

Statistics at a glance

The number 45 has the prime factorization:

$\!45=3^{2}\cdot 5$

Quantity	Value	Explanation
Total number of groups up to isomorphism	2
Number of abelian groups (i.e., finite abelian groups) up to isomorphism	2	(number of abelian groups of order $3^{2}$) times (number of abelian groups of order $5^{1}$) = (number of unordered integer partitions of 2) times (number of unordered integer partitions of 1) = $2\times 1=2$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.

The list

Group	GAP ID (second part)	Abelian?
cyclic group:Z45	1	Yes
direct product of Z15 and Z3	2	Yes