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 $|a^k|$?

Let $a in G$ such that $|a|=n$. Then,
a^n = a^{mk} = (a^k)^m = e_G

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:


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


Would highly appreciate an answer that a beginner would understand

differential geometry – Transitive action of a lie group in a connected manifold

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.

social media – Signals groups settings does not provide members option to hide their contact information from rest of the group members

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best practice – When displaying an “OK/Cancel” button group that is right-aligned should the main action be on the right?

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.

What should be the best practices concerning this situation?

Thanks in advance!

Adding taxonomy term to a group

I am trying to add taxonomy terms to groups (using the group module). I need terms to filter groups within a view. But as a result, the exposed filters don’t seem to work. There is no error, but the behavior of filters is not expected. When I request the display of A-term groups and B-term groups (from two different vocabularies), the view displays nothing. It is not logic.
Question: is group module compatible with taxonomy core module ? I mean: is it possible to add taxonomy terms directly to groups (and not group contents) ?

windows 10 – Will giving permissions and ownership to the administrator group also give permission to administrators on a different computer?

Say I set a file to have full control permission and ownership for the Administrators, if I transfer this to another computer with a different device name, will the file also have full control and ownership for administrators on that computer? If not, will changing the device name to the same as the original computer, make this the case? I ask because the administrator group has the device name before it, like this.

localization – Multilingual PNP provisioning for SharePoint Group

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:Localization LCID=”1034″ Name=”Spanish” ResourceFile=”SiteColumns-fr-FR.resx”/>
<pnp:Localization LCID=”1033″ Name=”English” ResourceFile=”SiteColumns-En-EN.resx”/>
<pnp:Templates ID=”CONTAINER-SharePointPalsDemo”>
<pnp:ProvisioningTemplate ID=”SharePointPalsDemo” Version=”1″ xmlns:pnp=”http://schemas.dev.office.com/PnP/2015/12/ProvisioningSchema”>


            <Field ID="{793F0419-9A72-48D9-B983-91BD359E5387}" Name="{resource:PNPFieldInternalName}" StaticName="MyCustomPNPField" DisplayName="{resource:PNPFieldDisplayName}" Type="Note" Required="FALSE" RichText="FALSE" Group="My Custom PNP Fields" />



            <!-- Bookmark ContentTypes -->
            <pnp:ContentType ID="0x010500C562BA980E1C457B97298CA27EC2C843" Name="MyCustomContentType" Description="Test Description" Group="Custom Group">
                    <pnp:FieldRef ID="793F0419-9A72-48D9-B983-91BD359E5387" Name="{resource:PNPFieldInternalName}" />


                        <pnp:User Name="{resource:Owner}"></pnp:User>


                    <pnp:RoleAssignment Principal="{resource:GroupName}" RoleDefinition="Contribute">



7 – OG Malinglist rejecting basic email sent to Organic Group

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!