Let . Show that G is the cyclic group of order six.
To show that \( G \) is the cyclic group of order six, we need to find an element of order six in \( G \). In the given presentation of \( G \), we have \( a^2 = e \), \( b^3 = e \), and \( aba^{-1}b^{-1} = e \), where \( e _____ ____ _____ ____ _____ ______ _________ ___ _______ _________.
________ __________ ___ ________ ____ ________ ______ _____.
_____ ___ ________ _______ _____ _____.
_________ ___ _____ _______ _______ _______ ___ _________ __________ _____ ___.
___ __________ __________ ________ ____ __________ _______ __________ _________ _______ ____.
______ __________ ________ _____ ________ ____ _____ ___ _____.
______ ____ ________ ____ _______ _______ _______ ___.
______ _____ ___ ________ _________.
___ _______ _________ ______ ___ __________ ________ ______ ____ _______ _____.
________ ________ _______ _______ _______ ________ ____ ____.
______ _____ ___ _______ _________ _____ _____ ______ ____ _______ _________.
________ ________ ___ ____ __________ ________ _________.
_________ _____ ________ _________ ____ _____ ____ ______.
_____ _________ ___ _____ __________ _____ ______ _______.
_______ ____ ____ _________ _______ _________.
______ _____ ________ ______ ________ _____ ______ ________ ___ __________ _________ _______.
______ ______ ____ ______ ________.
_______ ________ ____ _______ ______ _____ ____.
____ ______ ______ __________ _______ __________ ________ ____.
__________ ___ __________ ______ ______ _____ ___ ______ ______ ________ ___ __________.
_________ _______ __________ ___ ________ _______ ________ ________ ____ ________.
____ _________ ______ ___ ___ ______ ____ __________ _________.
_______ _________ _______ __________ ______ _________ _____ _____ ________ _______.
_______ ____ _______ _____ ________ ____.
_________ __________ ____ _______ __________ _________ ___ ________.
_____ ____ ___ _____ ________ ___ ____ ________ __________ __________.
________ _________ __________ ______ ___.
________ _____ _____ ________ _______ ________ ________ __________ ____ _________ _________ __________.
_______ ___ ____ ____ ________ ___ ____ _____ _____ ______ ______ ________.
_____ _______ ________ _________ _________.
____ ___ _________ __________ _______ ________ ____ _____.
_____ _____ ________ ________ ________ ________ _______ _____ ___ __________.
Get Full Answer on WhatsApp
By looking at the factorisation of guess the number of irreducible polynomials of degree 2 over
. Find all the irreducible polynomials of degree 2 over
.
Determine, up to isomorphism, all the finite groups with exactly 2 conjugacy classes
Let . Show that G is the cyclic group of order six.
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 ?, then for all
.
(b) There are exactly 8 elements of order 3 in .
(c) If , then
.
(d) .
(e) For any
Consider the natural action of on
, the set of
real matrices, by left multiplication.
(i) Under this action, if , show that the stabiliser of
is {I}, where I is the 2 × 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 . So suppose x ≠ 0. Let us write
. Then,
for non-zero
. Why ?
(iii) Let be a vector that is not a scalar multiple of
. Show that there is a matrix
such that
and
.(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 for which I − b is invertible.
Show that is not a UFD by giving two different factorisations of 20
The aim of this exercise is to show that SP2(R)acts transitively on R2 \ {0}.