Find the elementary divisors and invariant factors of
To find the elementary divisors and invariant factors of the abelian group \( G = \mathbb{Z}_8 \times \mathbb{Z}_{12} \times \mathbb{Z}_{15} \), follow these steps: --- ### Step 1: Prime Factorization of Orders Each cyclic group in \( G \) has an order: \[ 8 = 2^3, \quad 12 = 2^2 \times 3, \quad 15 = 3 \times 5 \] --- ### ______ ____ ________ ________ _______ _________ ________ _________.
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Solve the following set of congruences:
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.
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.
Show that
(i) Show that a matrix is symplectic if and only if ad - bc = 1.
(ii) Show that, to prove that SP2 acts transitively on GL2
, it is enough to show that, for any vector
, there is a 2 x
Which of the following statements are true and which are false? Give reasons for your answer.
(a) If a finite group G acts on a finite set S, then Gs1 = Gs2 for all s1, $2 ∈ X.
(b) There are exactly 8 elements of order 3 in S4.
(c)
(d)
(e) For any
Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.
Find all the Sylow 2-subgroups of Dn, when n is odd. Describe them in terms of x and y.