theory – Units in the group ring over fours group after Gardam

Giles Gardam recently found that Kaplansky’s unit conjecture fails on a virtually abelian torsion-free group: Consider the fours group $P = langle a, b ;|; (a^2)^b = a^{-2}, (b^2)^a = b^{-2} rangle$ (Gardam calls it the Promislow group). This group is a join of two Klein bottle groups and fits in a non-split exact sequence $1 rightarrow mathbb{Z}^3 rightarrow P rightarrow mathbb{Z}_2^2 rightarrow 1$. Now in the group ring $mathbb{F}_2(G)$ we have $pq = 1$ where
$$ p = 1 + aa + aaa + aab + AABA + aaBA + aaBABA + aabABA + aabb + ab + aba + abaBa + ababa + aBAbb + Abb + b + bA + BABA + bABA + BAbb + bb $$
$$ q = 1 + a + AA + aaab + aaababb + AAAbb + aaB + AABB + Ab + aB + AbaB + Abab + abaB + abab + AbaBa + Ababa + abaBB + abbb + B + BA + BB, $$
so Kaplansky’s unit conjecture is false.

So an the obvious question is:

What can we now say about the group of units in $mathbb{F}_2(G)$?

I claim no originality in observing that this question can be asked (I saw Avinoam Mann discuss this question on twitter). I gave this five minutes of thought myself and I see that the group is residually finite and contains a copy of $P$ (:P). Two random questions that are particularly interesting to me (but any information is welcome).

Is this group finitely-generated? Is it amenable?

Such questions do not seem obviously impossible on a quick look (for the first, some kind of Gaussian elimination? for the latter, use the fact the group has very slow growth somehow?), but they don’t seem obvious either. I assume there is no information in the literature explicitly about the units in group rings of torsion-free groups, because we didn’t know they can be nontrivial. But people have studied group rings a lot and I don’t have much background on them, so maybe more is known than I was able to see. One fun thing to look at is the following:

What is the isomorphism type of the subgroup $langle p, P rangle$ of the group of units?

I checked that p has order at least $10$, and I suppose it must have infinite order. I checked also that the $3$-ball generated by $p$ and $a$ has no identities, but $p$ and $b$ satisfy some identities.

MySQL group replication – how to find queries causing replication lag?

How can we determine which queries are causing a set of MySQL servers running Group Replication to have replication lag?

We have a MySQL Group Replication cluster with 9 servers, and we are currently experiencing replication lag (i.e high values in performance_schema.replication_group_member_stats.COUNT_TRANSACTIONS_REMOTE_IN_APPLIER_QUEUE for several servers).

We noticed that neither slow queries nor frequent queries are necessarily correlated with lag across servers. We have also noticed that some specific joins we had in our application were causing lag spikes and, once we removed them, the lag reduced. This is, however, not the case for all joins.

Is there a tool or a methodology we can use to find the queries causing the lag?

Stop Instagram Group Message Requests Spam

Instagram has a huge spam problem.

My “Message Requests” inbox is continually filled with requests to join Group Messages, sent by fake “sexy women” profiles.

The only options are to Accept, Ignore, or Delete these group message requests. There does not seem to be a way to report them as spam, or block the sender.

enter image description here

Message Requests
Requests aren’t marked as seen until you allow them.
Accept | Ignore | Delete
Delete All

How can I prevent these messages from being sent to me? How can I block these senders or report them as spam?

MySQL Group Replication starts master node with super_read_only

I am trying to set up MySQL group replication. Only problem is, that when I try to start replication group, It starts with super_read_only.

Here the configurations in my.cnf file


max_binlog_size = 4096
default_authentication_plugin     = mysql_native_password

log_bin                           = mysql-bin-1.log
enforce_gtid_consistency          = ON
gtid_mode                         = ON
log_slave_updates                 = ON
binlog_checksum                   = NONE

plugin-load-add                   =
plugin-load-add                   =
relay_log_recovery                = ON
transaction_write_set_extraction  = XXHASH64
loose_group_replication_start_on_boot                    = OFF
loose_group_replication_group_name                       = 74fe8890-679f-4e93-9169-a7edfbc1d427
loose_group_replication_group_seeds                      = mysql_cluster_mysql0_1:3306, mysql_cluster_mysql1_1:3306, mysql_cluster_mysql2_1:3306
loose_group_replication_single_primary_mode              = ON
loose_group_replication_enforce_update_everywhere_checks = OFF
bind-address =

instances are run inside docker, that’s why group seed addresses has these hostnames.

Also here the procedure for running master instance.


USE `db`$$

DROP PROCEDURE IF EXISTS `set_as_master`$$

CREATE DEFINER=`root`@`%` PROCEDURE `set_as_master`()
  SET @@GLOBAL.group_replication_bootstrap_group=1;
  CHANGE MASTER TO MASTER_USER='root' FOR CHANNEL 'group_replication_recovery';
  -- SELECT * FROM performance_schema.replication_group_members;


After running CALL start_as_master; in Sqlyog, process stucks on below lines.

'CHANGE MASTER TO FOR CHANNEL 'group_replication_recovery' executed'. Previous state master_host='', master_port= 3306, master_log_file='', master_log_pos= 4, master_bind=''. New state master_host='', master_port= 3306, master_log_file='', master_log_pos= 4, master_bind=''.

2021-03-03T21:47:55.934818Z 8 (System) (MY-013587) (Repl) Plugin group_replication reported: 'Plugin 'group_replication' is starting.'

2021-03-03T21:47:55.935929Z 9 (System) (MY-011565) (Repl) Plugin group_replication reported: 'Setting super_read_only=ON.'

Why does it run with super_read_only=ON?
Is there anything I miss during configuration or running script?

MySQL version is 8.0.23.

Is quotient of projective scheme over arbitrary base by a finite group also projective

This question probably follows from standard geometric invariant theory. If true I’d to know a reference for it. Given a projective scheme $Xrightarrow S$ over the base $S$. Let’s assume a finite group $G$ is acting on $X$ and its quotient is an $S$-scheme $X//G$. Is the quotient projective or at least proper? (I have seen versions of this over fields but not for arbitrary base.)

how to group export order by product in file csv in magento 2.3

j’ai une command export:order qui permet d’exporter les commandes dans un fichier csv et il y a un problème d’exportation de donnée en groupe par produit
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How to check if user belongs to a sharepoint AD group nested within a SPO group using REST/Javascript only?

I need to know if a user belongs to a specific SPO group using REST/Javascript. If the user has been added individually I can but if the user has been added to that SPO group via an AD group I can’t because rest APIs return the AD group as user hiding all its members.

I can’t use the workaround described here because I need that everyone could view group membership.

Any suggestion will be highly appreciated.

Best UI pattern to group multiple uncategorised items (i.e. questions)

Problem: User has an ability to add x number of questions (uncategorized questions). Normally, user adds 2 – 5 questions. However, the problem is when the user add more than 10. The tricky part that I haven’t thought of how to implement is when they add 200 questions.

The current implementation was just to stack up together the questions so user can easily make some changes to all questions. However, if that is more than 10 question, user needs to scroll down which is tedious but still works. But with 200 questions, that’s like unlimited scrolling.

enter image description here

Some solutions I thought:

This definitely is not a good solution. It’s confusing. I was thinking maybe there’s a good way to make use of pagination though so maybe you have some ideas.

Checkbox Table
I think this approach could be possible. The problem though is user can’t easily make some changes on the question. In addition, that means I would have another functionality of adding groups. There are possible workflow changes to suit this kind of approach but can’t think of any yet at the moment.

Note: The user can select the type of question he wants to add: multiple choice or essay-type.

I’m currently exploring on the best solution and to be honest, I haven’t really thought of any design approach to take yet. Would love to know our thoughts if you encountered something similar.

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Any one know how to destroy or End the Facebook Group |

In My Country there are lots of Scammers who eaten my money
They creates the fake currency exchange to eats simple person hard work money
Any one know how to destroy forever Facebook group or fb profile
I losts 1000$ of to exchange my money
So Webmaster please suggest


sharepoint online – How to EnsureUser with AD group in REST

I am trying to add AD group to SharePoint site in REST using EnsureUser but I kept getting error message “The specified user c:0+.w|S-1-5-21-1814438218-152777602-930774774-762833 could not be found“. Can someone please tell me what the correct format is for the logonName string? AD is on prem and we are using SP Online. Below is the code:

        string digest = formDigest == "" ? GetRequestDigest() : formDigest;
        string logonNameStr = WebUtility.HtmlEncode("c:0+.w|") + "S-1-5-21-1814438218-152777602-930774774-762833"; <== This seems to be incorrect
        string ensureUserUrl = _siteUrl + "_api/web/EnsureUser"; 
        using (var handler = GetHttpClientHandler(_siteUrl))
            using (HttpClient client = GetHttpClient(handler))
                client.DefaultRequestHeaders.Add("X-RequestDigest", digest);
                var payload = new { logonName = logonNameStr };
                string jsonBody = JsonSerializer.Serialize(payload);
                using (StringContent content = new StringContent(jsonBody))
                    content.Headers.ContentType = MediaTypeHeaderValue.Parse("application/json;odata=verbose;charset=utf-8");
                    HttpResponseMessage response = client.PostAsync(ensureUserUrl, content).Result;
                    string jsonStr = await response.Content.ReadAsStringAsync();
                    if (!response.IsSuccessStatusCode) //if adding user failed
                        throw new Exception(String.Format("Failed to add AD group {0} to SharePoint site: {1}", groupName, jsonStr));