Algebra 2 - Mid Sem (Sem 2)
Each of the 6 questions is worth 20% of the marks
1. a) Define the action of a group G on a
set S
b) If the action is
transitive, show that the isotropy groups at any 2 points are isomorphic.
c) Let G be any group.
Show that G x G acts on G if we set (s,t)g = sgt -1 for all s, t, g
in G. Show that the isotropy group at any point is isomorphic to G and determine
it as a subgroup of G x G.
2. a) For any prime number p, define a p-sylow
subgroup of a finite group.
b) Let G be a group
and N a normal subgroup of G. Show that any p-sylow subgroup of G maps
surjectively on to a p-sylow subgroup of G/N under the natural map G
→ G/N
c)
Give a 2-sylow subgroup of the alternating group A6.
3. a)
Determine the group of automorphisms of
b)
Consider the semi-direct product of G with
for the natural action.
Give a 3-sylow subgroup of this semi-direct product.
4. a) Show
that there exists a non-degenerate alternating bilinear form Rn
x Rn → R, if n is even.
b)
If B is a non-degenerate alternating bilinear form on V and and non-zero vector
v in V, show that there exists a vector w such that B restrics to the vector
space spanned by v and w as a non-degenerate form.
c)
Retaining the same notation, if dimension of V is greater than 2, show that
there exists a 2-dimensional subspace of V to which B restricts to the zero
form.
5. a)
Define the signature of a non-degenerate symmetric bilinear form on Rn.
b)
Prove that the signature is well defined.
c)
Show that the quadratic form
of the
real vector space of (2 X 2) matrices is non-degenerate and compute its
signature.
6. a)
Define projective space associated to a vector space.
b)
Show that the natural action of GL(n,R) on Rn induces
an action of GN(n,R) on Pn-1
c)
Let V be a vector space of dimension n over the field Fq consisting
of q elements. Calculate the cardinality of P(V)