Math 520A Written Assignment No. 1

due Monday, February 14, 2005

Directions. It is intended that you work these as exercises. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises either in books or in online references should not be required and is undesirable. If you make use of a reference other than class notes, you must properly cite that use.

You may not seek help from others.


Notation: Let F be a field. The following notations will be used.

F^{*} the multiplicative group of F
GL_{n}(F)the group of all n \times n invertible matrices in F
detthe homomorphism GL_{n}(F) --> F^{*} given by taking the determinant of a matrix
SL_{n}(F)the kernel of the homomorphism det
nu_{n}the homomorphism F^{*} --> GL_{n}(F) given by a :--> a · 1_{n} , (1_{n} the identity)
PGL_{n}(F)the quotient group GL_{n}(F)/Im(nu_{n})


  1. Find the order of the group GL_{n}(F) of invertible n \times n matrices with entries in a field F having q elements.

  2. Find a subgroup H of GL_{n}(F) such that GL_{n}(F) is isomorphic to the semi-direct product of H with SL_{n}(F) for the action of the former (by conjugation within GL_{n}(F)) on the latter.

  3. Let F be a field, and let alpha denote the action by fractional linear transformations of GL_{2}(F) on the set F cup {INFTY}.

    1. Show that alpha is a transitive action.

    2. What is the isotropy group at INFTY?

    3. Explain briefly why alpha is induced by an action of PGL_{2}(F).

  4. Let F_{3} denote the field of 3 elements. Observe that the order of the group GL_{2}(F_{3}) is 48 and that the groups SL_{2}(F_{3}) and PGL_{2}(F_{3}) both have order 24. The group S_{4} of all permutations of a set of 4 elements also has order 24. Determine which, if any, of these three groups of order 24 are isomorphic.

  5. Let F be a field, and let V be a finite-dimensional vector space over F. V^{*} will denote the dual space of V. The Heisenberg group Hs(V) is the set V \times V^{*} \times F with group law given by

     (v_{1}, f_{1}, t_{1}) * (v_{2}, f_{2}, t_{2})   =    (v_{1} + v_{2},  f_{1} + f_{2},  t_{1} + t_{2} + f_{2}(v_{1}))  . 

    1. Show that the center C of Hs(V) is the subgroup {0} \times {0} \times F.

    2. Explain why there is no subgroup H of Hs(V) such that Hs(V) is a semi-direct product of H and C.

    3. What action by automorphisms of V^{*} on the direct product V \times F gives rise to a semi-direct product of V^{*} and V \times F that is isomorphic to Hs(V)?