Talk:Binary tetrahedral group

binary polyhedral groups

If I've understood this right, all the binary polyhedral groups (i.e. binary cyclic, b. dicyclic, b. tetrahedral, b. octahedral, b. Icosahedral) have exactly one element of order 2. If I am right, this seems to me (A) probably connected with the name "binary", and (B) worth mentioning in the article. Maproom (talk) 11:40, 14 February 2008 (UTC)

This is correct since all the binary polyhedral groups are subgroups of the unit quaternions containing −1 which is the unique involution in that group. I believe, however, that the adjective "binary" comes from the fact that these groups are double covers of the ordinary polyhedral groups and not from the presence of a unique involution. -- Fropuff (talk) 22:18, 14 February 2008 (UTC)

False statement in first paragraph

The first paragraph includes the statement:

"It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 24."

But the octahedral group itself is order 24, and the binary octahedral group is order 48.Daqu (talk) 21:01, 16 October 2012 (UTC)

It looked to me like a typo for "tetrahedral", so I've fixed it. Maproom (talk) 08:09, 17 October 2012 (UTC)

Automorphism Group or isomorphism to SL(2,3) seems wrong

In the section Properties it is claimed that the automorphism group is T, of order 12. However it says lower that 2T is ismorphic to SL(2,3). Calculations in GAP show that the automorphism group of SL(2,3) is of order 24, actually being S_4, so this cannot be the case.

┌───────┐   GAP, Version 4.6.4 of 04-May-2013 (free software, GPL)
│  GAP  │   http://www.gap-system.org
└───────┘   Architecture: i686-pc-linux-gnu-gcc-default32
Libs used:  gmp, readline
Loading the library and packages ...
....

gap> Size(AutomorphismGroup(SL(2,3))); 24 gap>

I am not enough of a group theorist to get to the bottom of this, perhaps someone can check which of these statements is wrong. Timesuptim (talk) 08:11, 29 January 2015 (UTC)

Groupprops confirms your suspicion that the article is mistaken. Maproom (talk) 09:29, 29 January 2015 (UTC)

The "hat" symbol in ${\widehat {A_{4}}}$...what is the exact meaning?

Does that symbol denote double covering? I looked in List of mathematical symbols by subject and I did not find such mention there. I also looked in Double covering article and nowhere is it explained. I would like to see that explained, and referenced. --TheBlueWizard (talk) 06:38, 18 January 2018 (UTC)

Presentation: t-generator appears incorrect?

In the Presentation section, if t = (1/2)(1 + i + j - k) then the quaternion product r*s*t = (0, 0, -1, 0) while

r*t*s = (-1, 0, 0, 0), 

written by convention as "-1", is the desired answer. In order to match the conventions of the other Platonic binary groups, one should have instead t = (1/2)(1 + i - j + k) , and then r^2 = s^3 = t^3 = r*s*t = -1 as it was supposed to be, but got messed up?

It is also curious that the last line of Presentation omits "r s t = -1", while it is included in the first line, but is not correct, since with the current "t" none of the other powers is equal to r s t.

If the same person is maintaining the Binary Icosahedral page, note that the last line not only omits the 180-degree edge-flip quaternion

     r = (1/2)(0 + i/phi + j phi + k)
     r =   (1/2) {0, \tfrac{1}{\varphi},    \varphi,  1}

but also this final line is omitted that appears in every Binary group Wiki but the icosahedral:

  r^2 = s^3 = t^5 = r s t = - 1  

.

2600:1700:165E:8030:4D9A:A587:34B8:528D (talk) 03:46, 24 May 2023 (UTC)