Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.
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Let H be a finite group and, for some prime p, let P be a p-Sylow subgroup of H which is normal in H. Suppose H is normal in K, where K is a finite group. Then, show that Pis normal in K.
Compute the following:
(i) (ii)
Is there a finite group with class equation 1+1+2+2+2+2+2+2?
Let ? (?) be a finite extension F of odd degree(greater than 1). Show that ? (?2) = ? (?).
Consider the natural action of GL2 on M2
, the set of 2 x 2 real matrices, by left multiplication.
(i) Under this action, if det(x) ≠ 0, show that the stabiliser of x ∈ M2 is {I}, where I is the 2 x 2 identity matrix.
(ii) Suppose that det(x) = 0 in the remaining parts of this exercise. We will show that the stabiliser of x is infinite. If x = 0, the stabiliser of x is GL2. So suppose x ≠ 0. Let us write
Then,
for non-zero λ ∈ R. Why?
(iii) Let be a vector that is not a scalar multiple of
. Show that there is a matrix b =
such that b
= 0 and b
= α
(Hint: Set up two sets of simultaneous equations in two unknowns and argue why they have a solution.)
(iv) Check that I-b is in the stabiliser of x. Also, show that there are infinitely many choices of a for which I - b is invertible.
Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.
Describe the set of primes p for which x² - 11 splits into linear factors over Zp.