Determine whether each of the following is a group, i) N under addition. ii) Q[2

Question

Determine whether each of the following is a group, i) N under addition. ii) Q[2] {0} under multiplication. (Recall that Q[2] = {a + b 2: a, b epsilon Q}. AB = {x epsilon A: x B}.) iii) The set of two-by-two invertible real matrices under multiplication. iv) 3Z = {3z: z epsilon Z} under addition. v) 3Z{0} under multiplication. vi) R^3 under vector addition. vii) The set of all one-to-one correspondences from {1, 2, 3, 4, 5} to {1, 2, 3, 4, 5} under functional composition. (That is, the operation is degree, where (f degree g)(x) = J(g(x)).) E) Is T= {z epsilon C: |z| = 1} aring?

Explanation / Answer

i) Not group

there is no identity

ii) true

identity = 1

inverse for (a,b) = (a, -b) /a^2-2b^2

closure is there

associativity is there

iii) True

closure property A* B = C is also invertible C^-1 = B^-1A^-1

Identity (I) is there

iv) true

identity = 0

inverse =-x

v)false

idenity = 1

inverse doesn't exist

vi) true

(0,0,0) identity

(-x,-y,-z) inverse

vii) it is the permutation group

E) False

identity for addition not there

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