Talk:Linear group/Archive 1

Archive 1

Goals

I have listed below some ambitious goals for this page but don't have the mathematical foundation to realize them. Any help would be greatly appreciated.

First it seems that some of the definitions of the classical groups only consider the case where the field is R or C. I've left some comments on the talk pages for these groups so hopefully someone there will generalize the definitions.

Second it would be nice to provide short definitions for each of the types of classical groups listed on the page.

Third I believe the list can be increased to include "combinations" of these groups, such as the special orthogonal group and projective special orthogonal group. Unfortunately I'm not even sure which definitions make sense -- for instance I believe the symplectic group has a trivial center so it wouldn't make sense to talk about the projective symplectic group, but I'm not really sure about this. In short if someone could verify which of these group definitions are meaningful it would be very helpful.

Classical Groups

It would be nice to give short definitions of these groups and specify when they are simple (IE for which n and F)

• GL(n,F) - group is well covered in general linear group
• U(n,F) - definition of unitary group only uses C (left request on talk page)
• Sp(n,F) - definition of symplectic matrix appears general enough to cover arbitrary fields
• O(n,F) - The orthogonal group page gives a definition over arbitrary fields. Could use some examples over finite fields.

Special Linear Groups

When are these groups simple? When is the center nontrivial?

• SL(n,F) - group is well covered in general linear group
• SU(n,F) - definition of special unitary group only uses C (left request on talk page)
• SSp(n,F) - Sp is already a subgroup of SL so SSp = Sp
• SO(n,F) - The orthogonal group page gives a definition over arbitrary fields. Could use some examples over finite fields.

Projective Groups

When are these groups simple?

• PGL(n,F) - group is covered in projective linear group. Definition could be made more clear on how PGL is a matrix group.
• PU(n,F) - definition of projective unitary group only uses C
• PSp(n,F) - projective symplectic group needs to be written
• PO(n,F) - projective orthogonal group needs to be written

Projective Special Linear Groups

When are these groups simple?

• PSL(n,F) - group is covered in projective linear group. Definition could be made more clear on how PGL is a matrix group.
• PSU(n,F) - projective special unitary group needs to be written
• PSSp(n,F) - As mentioned above Sp is a subgroup of SL
• PSO(n,F) - projective special orthogonal group needs to be written

— Preceding unsigned comment added by TooMuchMath (talk • contribs) 17:30, 21 April 2006 (UTC)

question

what's meant by rare in " $GO_{n}(R)$ (rare) "? thanks. Mct mht 05:53, 22 April 2006 (UTC)

"uncommon". I deleted it since it seems to be confusing. R.e.b. 14:29, 22 April 2006 (UTC)