Jim got 2 out of 5 questions correct on his test. Which is equivalent to 2/5?
1 answer:
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Answer:
(a) No. (b)Yes. (c)Yes. (d)Yes.
Step-by-step explanation:
(a) If is an homomorphism, then it must hold
that ,
but the last statement is true if and only if G is abelian.
(b) Since G is abelian, it holds that
which tells us that is a homorphism. The kernel of
is the set of elements g in G such that . However,
is not necessarily 1-1 or onto, if
and
n=3, we have
(c) If remeber that
, which tells us that
is a
homomorphism. In this case
, if we write a
complex number as , then
, which tells
us that is the unit circle. Moreover, since
the mapping is not 1-1, also if we take a negative
real number, it is not in the image of , which tells us that
is not surjective.
(d) Remember that , using this, it holds that
which tells us that is a homomorphism. By computing we see
that and
is the unit circle, hence
is neither injective nor
surjective.