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 for all
.
(b) There are exactly 8 elements of order 3 in S4.
(c) If , then
.
(d) .
(e) For any ,
.
(a) True. By the Orbit-Stabilizer Theorem, if a finite group (G) acts on a finite set (S), then (|G| = |G_s| \cdot |O_s|) for any (s \in S), where (G_s) is the stabilizer of (s) and (O_s) __________ ______ ________ _______ ______ ___.
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Find all the Sylow 2-subgroups of Dn when n is odd. Describe them in terms of x and y.
If , show that a polynomial ax2 + bx + c is irreducible iff
where
is the group of squares in
Show that (i) Show that a matrix
is symplectic if and only if
.
(ii) Show that, to prove that
acts transitively on
, it is enough to show that, for any vector
, there is a
symplectic matrix with
as the first column. (Hint: For any matrix A, what is
?)
(iii) Complete the proof by showing that, given any non-zero vector
, there is always a non-zero vector
such that
is symplectic.
If is a finite field show that there is always an irreducible polynomial of the form x3 - x + a where
. (Hint: Show that
is not a surjective map.)
Find where
.
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 for all
.
(b) There are exactly 8 elements of order 3 in S4.
(c) If , then
.
(d) .
(e) For any ,
.
Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes.
(b) Is there a finite group with class equation 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2?
(c) Compute the following: a)
b)