If f: G to H is an isomorphism function of two groups Y then try the following

If f: G to H is an isomorphism function of two groups and then try the following
f ^ -1: H to G is an isomorphism

permutation: the product of two symmetrical groups that act on a function

Taking into account the rational function
$$
little
begin {align *}
f & (x_1, x_2, x_3; y_1, y_2, y_3) \
& = frac { left (1- frac {y_1} {x_1} right) left (1- frac {y_2} {x_1} right) left (1- frac {y_3} {x_1} right) left (1- frac {y_1} {x_2} right) left (1- frac {y_2} {x_2} right) left (1- frac {y_3} {x_2} right ) left (1- frac {y_1} {x_3} right) left (1- frac {y_2} {x_3} right) left (1- frac {y_3} {x_3} right)} { left (1- frac {x_2} {x_1} right) left (1- frac {x_3} {x_1} right) left (1- frac {x_3} {x_2} right) left (1- frac {y_2} {y_1} right) left (1- frac {y_3} {y_1} right) left (1- frac {y_3} {y_2} right)} \
end {align *}
$$

in $ 6 $ variables, I would like to calculate the sum
$$ sum_ {w in S_3 times S_3} w (f), $$
where the first copy of $ S_3 $ in $ w $ It is permuting the indices of the variables. $ x_1, x_2, x_3 $ while the second copy of $ S_3 $ It is permuting the indices of the variables. $ y_1, y_2, y_3 $.

F[x1_, x2_, x3_, y1_, y2_, y3_] : = ((1 - y1 / x1) (1 - y2 / x1) (1 - y3 / x1) (1 - y1 / x2) (1 - y2 / x2) (1 - y3 / x2) (1 - y1 / x3) ​​(1 - y2 / x3) (1 - y3 / x3)) / ((1 - x2 / x1) (1 - x3 / x1) (1 - x3 / x2) (1 - y2 / y1) ( 1 - y3 / y1) (1 - y3 / y2))

Google Groups footer messy email

How do you clean the footer of the Google Groups so that it does not contain any cited text or, if necessary, only the unsubscribe link?

at.algebraic topology – equally divided into an isomorphism of $ KR $ -groups

In section 3, p. 8 of the theory of strings on elliptical curves, affirm that for $ X $ compact with $ x_0 $ A fixed point of involution, the map. $ i: x_o rightarrow X $ it's equanimous and evenly divided, which according to them implies

$ KR ^ {j} (X) approx KR ^ {j} (X – {x_0 }) oplus KO ^ {j} (x_0) $

I do not understand what division equitably means here in relation to the functor $ KR ^ j (-) $ Apart from that you can simply make this separation or why this is so.

1) I would appreciate if someone could put more steps in the isomorphism. $ KR ^ {j} (X) approx KR ^ {j} (X – {x_0 }) oplus KO ^ {j} (x_0) $

2) Is there any reference where they explain in more detail what is meant by equitable division? I have not seen any mention of the term as it is being used here for any kind of equivalent cohomology theory.

Keep in mind that it seems to be $ KR $-theory with compact supports but I do not know if this is essential for the previous argument.

Thanks for your time

Theory of groups: permutation number of S4 as a product of two separate cycles each of length 2

There was a problem of finding out the number of permutations of order 2 in S4.

There are two cases.

Case 1

Single cycle permutation of length 2.

case-2

permutations of two separate cycles each of length 2.

For case 1 total number of permutations will be $ frac {4P2} {2} = 6 $

and these permutations are $ (1 2), (1 3), (1 4), (2 3), (2 4), (3 4) $

For case 2 total number of permutations will be $ frac {4P2} {2} times frac {2P2} {2} = 6 times1 = 6 $

But The permutations of two separate cycles each of length 2 are
$ (12) (34), (13) (24), (14) (23) $

this is clearly 3 and not 6.

I'm sure I'm making some kind of mistake finding the number of permutations in case-2.

Please help.

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abstract algebra: direct sum of $ n $ (infinite) isomorphic cyclic groups to direct the sum of $ n $ copies of $ mathbb {Z} $?

I am currently studying a bit of algebra and I am currently covering the various equivalent definitions of free abelian groups. However, to understand why these definitions are really equivalent, this question arose and I have not been able to resolve it on my own. The real question is:

Assume that I have $ n $ infinite cyclic groups $$ langle a_1 rangle, langle
a_2 rangle, …, langle a_n rangle $$

Do you have that? $$ bigoplus_ {i = 1} ^ n langle a_i rangle cong
bigoplus_ {i} ^ n mathbb {Z} (n text {copies of} mathbb {Z}) $$
?

If anyone has a recommendation for a good book on this subject, I would also appreciate any recommendation.

Thanks for any help.

python: division of a column of values ​​into 3 groups according to numerical ranges

I need to classify my dependent variable of numerical values ​​into 3 categories based on different ranges. If it is greater> 75 = 0, 50 – 74 = 2, <50 = 3. Please let me know how to group them as categorical values ​​and make an adjacent column along with my original column.

Group theory Cohomology of simple finite groups.

Suppose $ G $ Y $ H $ They are finite simple groups.
I think it's obvious that one expects the cohomology groups with integer coefficients of $ G $ Y $ H $ they are not isomorphic unless $ G cong H $. Is there any proof of this? Keep in mind that I do not want to calculate those cohomology groups (and that I know has not yet been done completely), just to show that they are not isomorphic.

mount: can open LUKS encyrpted partition but not explore volume groups

I am trying to restore GRUB on my SSD after accidentally overwriting it by installing a Linux operating system on another disk (do not the same SSD).
To try to repair grub, I am using a live version of the operating system distribution from a USB stick.

Once inside the live distribution, I can see the disk where the installation of my OS and the original GRUB reside:

Disk / dev / sda: 232.9 GiB, 250059350016 bytes, 488397168 sectors
Units: sectors of 1 * 512 = 512 bytes
Size of the sector (logical / physical): 512 bytes / 512 bytes
I / O size (minimum / optimal): 512 bytes / 512 bytes
Type of disk label: gpt
Disk identifier: 0A88BA3C-EF26-4291-AD41-EF9331042E6D

Type of End Sectors for Device Start
/ dev / sda1 40 409639 409600 200M EFI System
/ dev / sda2 409640 23845871 23436232 11.2G Apple HFS / HFS +
/ dev / sda3 23846912 24319999 473088 231M Linux file system
/ dev / sda4 24320000 488396799 464076800 221.3G Linux file system

/ dev / sda1 is the EFI partition that contains the EFI configuration for MacOS and my Linux distribution.
/ dev / sda2 it's a MacOS installation and it's not relevant to the problem here.
/ dev / sda3 contains information of efi and grub:

-rw-r - r-- 1 root root 186567 May 7, 2018 config-4.9.0-6-amd64
drwxr-xr-x 2 root root 1024 jun 16 2018 efi
drwxr-xr-x 5 root root 1024 April 18 23:12 grub
-rw-r - r-- 1 root root 28489619 Feb 23 15:59 initrd.img-4.9.0-6-amd64
drwx ------ 2 root-cause 12288 Jun 16 2018 lost + found
-rw-r - r-- 1 root root 3190138 May 7, 2018 System.map-4.9.0-6-amd64
-rw-r - r-- 1 root root 4224800 May 7, 2018 vmlinuz-4.9.0-6-amd64

/ dev / sda4 It is the partition encrypted with user data.

I used the following guides as reference:

https://ubuntuforums.org/showthread.php?t=2266650

https://stephentanner.com/restoring-grub-for-an-encrypted-lvm.html

I can open the encrypted partition:

$ cryptsetup luksOpen / dev / sda4 sda4_crypt
Enter the passphrase for / dev / sda4:
$ 

Enter the password and the encrypted drive is automatically mounted on / media / user / 475cac44-f48a-4d17-8659-611a06e1f961.

However, when I try to find volume groups:

$ vgscan
Reading all physical volumes. This may take a while ...
$ 

Nothing is returned.

The same happens when you try to find logical volumes:

$ sudo lvscan
$ 

I can see the contents of the partition mounted and encrypted:
$ ls -la / media / user / 475cac44-f48a-4d17-8659-611a06e1f961 /

total 112
drwxr-xr-x 22 root root 4096 July 4, 2018.
drwxr-x --- + 4 root root 80 April 18 08:57 ..
drwxr-xr-x 2 root root 4096 March 9 17:16 tray
drwxr-xr-x 2 root root 4096 Jun 16 2018 boot
drwxr-xr-x 5 root root 4096 Feb 20 20:46 dev
drwxr-xr-x 131 root root 12288 Apr 17 08:23 etc.
drwxr-xr-x 3 root root 4096 June 16, 2018 at home
lrwxrwxrwx 1 root root 29 Jun 16 2018 initrd.img -> boot / initrd.img-4.9.0-6-amd64
lrwxrwxrwx 1 root root 29 Jun 16 2018 initrd.img.old -> boot / initrd.img-4.9.0-6-amd64
drwxr-xr-x 18 root root 4096 April 13 12:20 lib
drwxr-xr-x 2 root root 4096 Jun 16 2018 lib64
drwx ------ 2 root root 16384 Jun 16 2018 lost + found
drwxr-xr-x 4 root root 4096 April 17 08:23 media
drwxr-xr-x 6 root root 4096 Nov 25 14:41 mnt
drwxr-xr-x 4 root root 4096 sep 8 2018 opt
drwxr-xr-x 2 root root 4096 February 23, 2018 proc
drwx ------ 12 root root 4096 April 17 01:35 root
drwxr-xr-x 3 root root 4096 Feb 20 20:46 run
drwxr-xr-x 2 root root 12288 April 17 03:30 sbin
drwxr-xr-x 2 root root 4096 April 28, 2018 srv
drwxr-xr-x 2 root root 4096 February 23, 2018 systems
drwxrwxrwt 8 root root 4096 April 17 08:23 tmp
drwxr-xr-x 10 root root 4096 jun 16 2018 usr
drwxr-xr-x 11 root root 4096 Jun 16 2018 var
lrwxrwxrwx 1 root root 26 Jun 16 2018 vmlinuz -> boot / vmlinuz-4.9.0-6-amd64
lrwxrwxrwx 1 root root Jun 26 2018 vmlinuz.old -> boot / vmlinuz-4.9.0-6-amd64

But without seeing / knowing the volume group, how can I mount the correct directories to repair grub?