$$\text i beg your pardon is a team under multiplication by modulo n=\1,2,4,7,8,11,13,14\$$then $|2|=4$, since

\beginalign*&2^1=2\\&2^2=4\\&2^3=8\\&2^4=16\mod15=1\\&\textSo |2|=4.\endalign*

However, ns don"t understand just how this works for teams that don"t have any kind of relation come modulo. Take it $(\ugandan-news.combbZ,+)$ for instance. If I want to uncover the stimulate of $3$, then I require to uncover $n:3^n$ is same to the identity, i m sorry in this situation is $0$.

You are watching: Order of an element in a group

I intend my question have the right to be summarized together follows:

Does the order of an aspect only make sense if we are managing groups dealing with modulo?

abstract-algebra group-theory

share

mention

monitor

inquiry Oct 13 "14 in ~ 18:08

Sujaan KunalanSujaan Kunalan

10k1616 gold badges4949 silver badges8585 bronze badges

$\endgroup$

include a comment |

## 3 answers 3

energetic oldest Votes

9

$\begingroup$

Yes, it makes sense. The bespeak of an aspect $g$ in some group is the the very least positive essence $n$ such that $g^n = 1$ (the identity of the group),

*if any kind of such $n$ exists*. If over there is no together $n$, climate the bespeak of $g$ is characterized to it is in $\infty$.

As provided in the comment through

Travis, you deserve to take a small permutation team to obtain an example. For instance, the permutation $(1,2,3,4)$ in the symmetric team $S_4$ of level $4$ (all permutations that the collection $\1,2,3,4\$) has actually order $4$. This is due to the fact that $$(1,2,3,4)^1 = (1,2,3,4)\neq 1,$$$$(1,2,3,4)^2 = (1,3)(2,4)\neq 1,$$$$(1,2,3,4)^3 = (1,4,3,2)\neq 1$$and$$(1,2,3,4)^4 = 1,$$so $4$ is the smallest power of $(1,2,3,4)$ that yields the identity.

For the additive group $\ugandan-news.combbZ$ of integers, every non-zero facet has boundless order. (Of course, here, we use additive notation, for this reason to calculation the stimulate of $g\in\ugandan-news.combbZ$, us are searching for the least positive creature $n$ such that $ng = 0$, if any. But, uneven $g = 0$, there is no together $n$, therefore the bespeak of $g$ is $\infty$.)

share

point out

monitor

edited Oct 13 "14 in ~ 18:47

answer Oct 13 "14 at 18:10

JamesJames

8,59722 gold badges1818 silver badges3030 bronze title

$\endgroup$

3

add a comment |

1

$\begingroup$

No, the concept makes feeling for all teams (at the very least all limited groups, anyway as infinite groups can have facets with unlimited order), and also its an interpretation is simply the one girlfriend give. (All multiplicative subgroups the $\ugandan-news.combbZ_n$, i.e., integers modulo $n$ are abelian, however not all groups are abelian.)

re-publishing

point out

monitor

reply Oct 13 "14 in ~ 18:13

Travis WillseTravis Willse

80.1k88 gold badges9191 silver badges195195 bronze badges

$\endgroup$

include a comment |

1

$\begingroup$

A team can have actually finite or infinite variety of elements. Once the group has finite variety of elements, we check out the the very least POSITIVE

*n*i.e.(n>0) such the

*g^n*provides the identification of the team (in situation of multiplication) or

*n*g*gives the identification (in situation of addition).Here

*Z*has an infinite variety of elements. There does not exist any

*n>0*because that which you achieve identity. Therefore

*Z*is of infinite order.

re-superstructure

cite

monitor

reply Oct 13 "14 in ~ 18:16

Shikha SafayaShikha Safaya

3355 bronze badges

$\endgroup$

include a comment |

## her Answer

Thanks for contributing an answer to ugandan-news.comematics ridge Exchange!

Please be sure to*answer the question*. Provide details and also share your research!

But *avoid* …

Use ugandan-news.comJax to layout equations. Ugandan-news.comJax reference.

To find out more, see our advice on writing an excellent answers.

Draft saved

Draft discarded

### Sign increase or log in

authorize up making use of Google

authorize up using Facebook

authorize up making use of Email and also Password

submit

### Post together a guest

surname

email Required, however never shown

### Post as a guest

surname

Required, yet never shown

short article Your answer Discard

By clicking “Post her Answer”, friend agree come our terms of service, privacy policy and also cookie plan

## Not the price you're feather for? Browse other questions tagged abstract-algebra group-theory or ask your own question.

Featured ~ above Meta

27 votes · comment · stats

Linked

2

Calculate the bespeak of the following group elements.

See more: The Huntress And The Hero ( Atalanta And Achilles Or Atalanta Ideas

connected

1

The bespeak of an facet in a group

2

various way's of approaching the bespeak of an facet

0

detect the bespeak of Each aspect in a team

0

about The bespeak Of An facet In A group

1

stimulate of each aspect in a symmetry group.

2

Calculating the order of an aspect in the team $U_27$

0

deserve to I main point an aspect of a group to an facet of a various group?

4

stimulate of an facet 250 in U(641)

0

exactly how to uncover the stimulate of elements?

hot Network inquiries much more hot concerns

question feed

subscribe to RSS

question feed To subscribe to this RSS feed, copy and also paste this URL into your RSS reader.

ugandan-news.comematics

company

stack Exchange Network

site style / logo © 2021 ridge Exchange Inc; user contributions licensed under cc by-sa. Rev2021.9.24.40299

ugandan-news.comematics ridge Exchange works ideal with JavaScript enabled

your privacy

By click “Accept all cookies”, girlfriend agree ridge Exchange can store cookies on your machine and disclose information in accordance v our Cookie Policy.