Let p: Z → Z be a group homomorphism given by the rule o (x) = 10 . x %3D for any a E Z. a. Compute o (-8). 9(-8) = b. Find an element in the kernel Ker () of the group homomorphism p. E Ker (4) c. Find an element in the range Ran () of the group homomorphism p. E Ran (4) d. Is p injective? ? e. Is p surjective? ? f. Is y an isomorphism? ?

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.8: Some Results On Finite Abelian Groups (optional)
Problem 14E: Suppose that the abelian group G can be written as the direct sum G=C22C3C3, where Cn is a cyclic...
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Let p: Z → Z be a group homomorphism given by the rule
o (x) = 10. x
for any x E Z.
a. Compute o (-8).
o(-8)
b. Find an element in the kernel Ker () of the group homomorphism p.
E Ker (e)
c. Find an element in the range Ran (w) of the group homomorphism p.
E Ran ()
d. Is p injective?
?
e. Is p surjective?
f. Is p an isomorphism?
Transcribed Image Text:Let p: Z → Z be a group homomorphism given by the rule o (x) = 10. x for any x E Z. a. Compute o (-8). o(-8) b. Find an element in the kernel Ker () of the group homomorphism p. E Ker (e) c. Find an element in the range Ran (w) of the group homomorphism p. E Ran () d. Is p injective? ? e. Is p surjective? f. Is p an isomorphism?
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