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 be a field. The following notations will be used.
the multiplicative group of | ||
the group of all invertible matrices in | ||
det | the homomorphism given by taking the determinant of a matrix | |
the kernel of the homomorphism det | ||
the homomorphism given by ( the identity) | ||
the quotient group |
Find the order of the group of invertible matrices with entries in a field having elements.
Find a subgroup of such that is isomorphic to the semi-direct product of with for the action of the former (by conjugation within ) on the latter.
Let be a field, and let denote the action by fractional linear transformations of on the set .
Show that is a transitive action.
What is the isotropy group at ?
Explain briefly why is induced by an action of .
Let denote the field of elements. Observe that the order of the group is and that the groups and both have order . The group of all permutations of a set of elements also has order . Determine which, if any, of these three groups of order are isomorphic.
Let be a field, and let be a finite-dimensional vector space over . will denote the dual space of . The Heisenberg group is the set with group law given by
Show that the center of is the subgroup .
Explain why there is no subgroup of such that is a semi-direct product of and .
What action by automorphisms of on the direct product gives rise to a semi-direct product of and that is isomorphic to ?