Let $(G,*)$ be a group and $a in G$. Suppose that $|a|=n$ and $n=mk$ for some positive integers $m$ and $k$. What is $|a^k|$?

Let $(G,*)$ be a group and $a in G$.
Suppose that $|a|=n$ and $n=mk$ for some positive integers $m$ and $k$.
What is $|a^k|$?

attempt:
Let $a in G$ such that $|a|=n$. Then,
begin{equation*}
a^n = a^{mk} = (a^k)^m = e_G
end{equation*}

where $e_G$ is the identity element of $G$. Thus, the order of $a^k$ is $m$.

In particularly, by theorem:
Let $(G,*)$ be a group, $a in G$, and
$|a|=n$. Then, for all $k in Bbb N$,
$|a^k| = frac{n}{gcd(k,n)}$,
we have, $|a^k| = frac{n}{gcd(k,mk)} = frac{n}{k} = m$.

Does it true? On the other hand, in the answer key says that the answer is
$frac{n}{m}$. How to get this?

R Circlize, basic chord with group sectors

I am trying to create a Chord diagram in Circlize with the following DF:

sampdat<-read.csv("H:\sample data.csv", header = T)

      targeta targetB source value
    1     AB1     AB2     AB   400
    2     AB2     AB3     AB   550
    3     CD1     CD2     CD   450
    4     CD2     CD3     CD   600

I run:

chordDiagram(sampdat)

to get

enter image description here

What I’m trying to achieve is to get the Chord to arrange the sectors according to “source”-column and also add a track that groups and covers all “A”-sectors, and “C”-sectors.

I’ve read some of the other similar questions here on the site but most answers use tons of code (some with loops, of which I don’t know how to use yet) without any in-depth explanation as to what it does so I’m struggling to adapt it to my needs. My understanding is that I’m somehow supposed to use

highlight.sector()

Would highly appreciate an answer that a beginner would understand

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I am trying to see why is it that if we have a transitive action of a lie group $Grightarrow M$ then if this action is transitive the connected component of the identity $G^0$ also acts transitively on $M$ and that for all $pin M$, $G/G_0 cong G_p/(G_pcap G^0)$.

Since the action is transitive we know that $G/G_pcong M$ and since the map $Grightarrow G_p$ is a submersion we get that the map $Grightarrow M, grightarrow g.p$ is open . I don’t know how useful this is since we only know that $G^0$ is a closed set I don’t think it has to be open since connected components don’t necessarily need to be open, and also I am not sure if this map is even closed or if it is how I could try and prove it.

Any enlightment is appreciated. Thanks in advance.

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I am designing an “OK/Cancel” button group on a modal hence this button group is right-aligned. Since this is the case I have placed the main action button “OK” on the right, and the secondary action button “Cancel” on the left.

On other pages these button groups are left aligned hence the main action “OK” button is on the left.

My colleague argues that “we should be consistent with our button order” regardless of the positioning of the button group.

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Did anybody try applying localization using PNP Provisioning for SharePoint groups? I have tried but it is not working. Can somebody please help?

”’
<pnp:Provisioning xmlns:pnp=”http://schemas.dev.office.com/PnP/2015/12/ProvisioningSchema”>
<pnp:Preferences Generator=”OfficeDevPnP.Core, Version=2.0.1601.0, Culture=neutral, PublicKeyToken=null” />
pnp:Localizations
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<pnp:Localization LCID=”1033″ Name=”English” ResourceFile=”SiteColumns-En-EN.resx”/>
</pnp:Localizations>
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        <pnp:SiteFields>


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    </pnp:ProvisioningTemplate>
</pnp:Templates>

</pnp:Provisioning>”’

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I am using OG Mailinglist 7.x-1.1-alpha2.
We have been using this module for years now.
We have a new user who is attempting to send emails to a couple of our lists which are the email part of Organic Groups which the module OG Mailinglist turns into listservs.
This new user is sending basic emails with no inline images or attachments and is getting each email sent rejected with the rejection response saying the emails could not be read properly.

Usually when this happens this is because of a bug with the library used to decode the emails in that the library malfunctions when you have both an attachment and an inline image in the body of the email, but the person having the issue is just sending text emails.

We would appreciate any insight.

Thank you!