# Group Theory: Finite Abelian Groups - An element of order

• Lelouch
In summary: You are on the right track! Try to find elements with order 45 in the groups you listed above. You should be able to find at least one element in each group with order 45. Remember that for an element ##a## to have order 45, you need ##a^{45} = e##, where ##e## is the identity element. Maybe try looking at the generators of the groups and see if you can find one that satisfies this condition.
Lelouch

## Homework Statement

Decide all abelian groups of order 675. Find an element of order 45 in each one of the groups, if it exists.

## Homework Equations

/propositions/definitions[/B]
Fundamental Theorem of Finite Abelian Groups
Lagrange's Theorem and its corollaries (not sure if helpful for this problem)

## The Attempt at a Solution

I used the Fundamental Theorem of Finite Abelian Groups to find the abelian groups. The prime factorization of 675 is
$$\begin{split} 675 &= 3 \cdot 3 \cdot 3 \cdot 5 \cdot 5 \\ & = 3^{2} \cdot 3 \cdot 5 \cdot 5 = 9 \cdot 3 \cdot 5 \cdot 5 \\ & = 3^{3} \cdot 5 \cdot 5 = 27 \cdot 5 \cdot 5 \\ &= 3 \cdot 3 \cdot 3 \cdot 5^{2} = 3 \cdot 3 \cdot 3 \cdot 25 \\ &= 3^{2} \cdot 3 \cdot 5^{2} = 9 \cdot 3 \cdot 25 \\ &= 3^{3} \cdot 5^{2} = 27 \cdot 25 .\\ \end{split}$$

and the groups are

$$\begin{split} \mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{5} \times \mathbb{Z}_{5} \quad & \land \quad \mathbb{Z}_{9} \times \mathbb{Z}_{3} \times \mathbb{Z}_{5} \times \mathbb{Z}_{5} \\ \mathbb{Z}_{27} \times \mathbb{Z}_{5} \times \mathbb{Z}_{5} \quad & \land \quad \mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{3} \times \mathbb{Z}_{25} \\ \mathbb{Z}_{9} \times \mathbb{Z}_{3} \times \mathbb{Z}_{25} \quad & \land \quad \mathbb{Z}_{27} \times \mathbb{Z}_{25} . \end{split}$$

I am stuck on the second question "Find an element of order 45 in each one of the groups, if it exists.". I know that I have to find an ##a \in G## (where G is each of the above abelian groups) such that ##order(a) := \#(<a>) = \#(\{k \cdot a : k \in \mathbb{Z}\}) = 45##, where ##\#(\bullet)## is the cardinality of a set; or show that such an element a does not exist.

Hints are very much appreciated.

Last edited:
You surely need a subgroup ##U## with ##45\,|\,|U|\,.## So what about subgroups which contain ##\mathbb{Z}_{45}\,?##

fresh_42 said:
You surely need a subgroup ##U## with ##45\,|\,|U|\,.## So what about subgroups which contain ##\mathbb{Z}_{45}\,?##

I don't quite understand. I am really lost on this one. If I am supposed to find a subgroup U with ##45 | |U|##, then this subgroup must have ##|U| = 45, 90, 135, ... ##

Given an element ##g## of order ##45## in ##G##, it must be part of a subgroup ##U## with at least ##\mathbb{Z}_{45} \subseteq U##. We have in addition ##|U|\,|\,|G|=675## so ##|U|=90## is not possible.

## What is a finite abelian group?

A finite abelian group is a group that has a finite number of elements and satisfies the commutative property, meaning that the order in which the elements are multiplied does not affect the result.

## What does it mean for an element to have order in a finite abelian group?

The order of an element in a finite abelian group is the smallest positive integer n such that when the element is multiplied by itself n times, the result is the identity element of the group. In other words, the order of an element is the number of times that it needs to be combined with itself to reach the identity element.

## Can an element have multiple orders in a finite abelian group?

No, an element can only have one order in a finite abelian group. This is because the order of an element is unique and is determined by the size of the group and the element's properties.

## How can the order of an element be used to classify finite abelian groups?

The order of an element can be used to classify finite abelian groups because it reveals information about the structure of the group. For example, if all elements in a group have the same order, the group is cyclic. If there are elements with different orders, the group is not cyclic but may still have some cyclic subgroups.

## Can the order of an element in a finite abelian group change?

No, the order of an element in a finite abelian group cannot change. This is because the order is an inherent property of the element and is determined by the structure of the group. Even if the element is combined with other elements, its order will remain the same.

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