I am reading “Lie Groups and Lie Algebras and some of Their Applications” by Robert Gilmore and I have become hopelessly confused over what I am sure is a simple matter. With $H$ a normal subgroup of the group $G$ Gilmore writes:

“A right coset is the set of group operations $c_0$, $c_1$, ... $\in G$ with the property that $H c_0 + H c_1 + ... = G$ and, furthermore, no element $g$ of $G$ is contained more than once in the sum on the left.”

First, is this a non-standard definition of a “coset”? I recall learning that a coset (not “the coset”) was a set of the form $Hg$ and that what Gilmore is describing is called a “transversal” of the cosets or a collection of “coset representatives.” He goes on to provide the following theorem:

“If $H$ is an invariant subgroup of $G$ then the coset elements $c_0$, $c_1$, $c_2$, ... can be chosen in such a way that they are closed under multiplication and form a group called the factor group $G/H$

So it seems to me that under Gilmore’s definition we speak of “the coset” of a group $G$ with normal subgroup $H$ instead of “the cosets” which, the way I recall learning it, are several sets that partition $G$ and which are each elements of the factor group. Furthermore I tried to find a multiplicatively closed collection $c_i$ for $D_4$ (dihedral group of order 8 - the symmetries of a square) with $H=\{I, R_{180}\}$ and I can’t do it. I was able to do it for $A_4$ with $H=\{(1), (12)(34), (13)(24), (14)(23)\}$. Am I just confusing myself somehow?

Gilmore later asks the reader to prove that a collection of coset representatives can always be chosen in such a way that they form the same group as $G/H$ when $H$ is a normal subgroup of $G$. Can someone give me a hint about how to do that proof and maybe demonstrate this for $D_4$?

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    $\begingroup$ His treatment sounds eccentric, to say the least. $\endgroup$ – Lord Shark the Unknown Feb 23 at 17:29
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    $\begingroup$ I couldn't believe it so I checked, but this is really written. Use another book, there are many serious ones. $\endgroup$ – YCor Feb 23 at 19:24
  • $\begingroup$ So....am I right that this is not possible for $D_4$? Is the theorem even correct? Is it correct always or just under certain circumstances. Isn’t the theorem equivalent to, “We can always find a transversal associated with the cosets of a normal subgroup such that the transversal is itself a group?” $\endgroup$ – BoundaryValue Feb 23 at 19:45
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    $\begingroup$ I think you are right. It would follow that every quotient of a group is isomorphic to a subgroup. $\endgroup$ – Berci Feb 23 at 19:53
  • $\begingroup$ Thank you YCor and Berci for your comments. I think I just need to accept that the book has a pretty basic error. I have answered my own question (see answer) and noted that Gilmore actually said what Berci said in a second book! $\endgroup$ – BoundaryValue Feb 25 at 15:01

To the point made by Berci, in Gilmore’s other book, “Lie Groups, Physics, and Geometry” Gilmore doubled down on all this and basically said what Berci said in a comment to my posted question except Gilmore seems to believe it is true. In that book Gilmore again defined a coset as the subset of G that will non-redundantly produce all of G when multiplied by all of the elements of H. He then assigned this “coset” the symbol “G/H”. As far as I can tell, that is so non-standard that it is actually wrong. Then Gilmore wrote,

“In general it is not possible to chose the group elements of G/H in such a way that they from a subgroup of G. However, if H is an invariant subgroup of G then it is always possible to choose the group elements in the quotient space G/H such that they form a subgroup in G. This group is called the factor group also denoted G/H. Since $A_3$ is an invariant subgroup of $S_3$ The coset $S_3 / A_3$ is a group and this group is isomorphic to $S_2$.”

I suppose it is time to conclude that despite being an excellent book overall, Gilmore is simply wrong about this for some strange reason. I feel weird about it, since this book is sort of the Lie Group Bible for physicists and this particular topic is so fundamental. Maybe that is the point: it is such an elementary topic that the detail slipped by him. Factor groups can be confusing and I can imagine that Gilmore’s misconception could remain unnoticed because the errant fact does not propagate throughout the subject of Lie Groups. I guess. I still feel like I am missing something.

I am answering my own question, which is rude. I still would like some more people to check the references (Gilmore’s books) and validate my conclusion here.


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