Decide whether each of the following sets is a subgroup of G = { 1 , − 1 , i , − i } under multiplication. If a set is not a subgroup, give a reason why is not. a. { 1 , − 1 } b. { 1 , i } c. { i , − i } d. { 1 , − i }
Decide whether each of the following sets is a subgroup of G = { 1 , − 1 , i , − i } under multiplication. If a set is not a subgroup, give a reason why is not. a. { 1 , − 1 } b. { 1 , i } c. { i , − i } d. { 1 , − i }
Solution Summary: The author explains the Theorem for Equivalent Set of Conditions for a Subgroup.
Decide whether each of the following sets is a subgroup of
G
=
{
1
,
−
1
,
i
,
−
i
}
under multiplication. If a set is not a subgroup, give a reason why is not.
Determine which of the following are groups under the stated operations. For those
which are groups, show that the axioms hold and state the identity and inverses.
For those which are not, show that one of the axioms fails.
i. The pair ({2, 4, 6, 8), o) where x oy
=
ii. The set {a +b√3|a, b = Z} under addition.
iii. The set of all vectors in R³ and the operation is vector product.
xy mod 10.
Show that the following are homomorphisms. Calculate their kernel and image,
showing your working. Giving reasons for your answers, which of the homomor-
phisms are injective and which are surjective?
i. y: 27 → Z7 given by (x) = 3x mod 7.
ii. p: G→ (R*, x) where G = =
{ (89) | a € P
ER}
plication and p(A) = det (A) for A E G. (Recall that R* = R\{0}.)
iii. Z→ S3 given by (n) = (123)" where S3 is the group of permutations
on 3 objects.
ER*,bER
with matrix multi-
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