8. Prove or disprove (a). There is a homomorphism between any two groups. (b). There is a finite group to an infinite group. (c). Any two finite groups the same order are isomorphic. (d). There is an abelian group isomorphic to a non-abelian group. (e). The map G defined by p(x) = x¹ is a homomorphism for any a group (G, *). (f). For any two groups (G, *) and ( *), we have Gx G'G'xG. (g). The map : (C, +) → defined by p(x + y) = x +y is an epimorphism. (h). There are 5 subgroups of 42/64Z under the usual addition. (i). Let (Z, +) be the group of integers. The map 4 : Z× Z → Z defined by (a, b) = a - b is a homomorphism and ker(4) = {(a, a): a € Z}. (j). Z/nZ Zn, for any positive integer n (under addition). Dr bratam Alaluboot
8. Prove or disprove (a). There is a homomorphism between any two groups. (b). There is a finite group to an infinite group. (c). Any two finite groups the same order are isomorphic. (d). There is an abelian group isomorphic to a non-abelian group. (e). The map G defined by p(x) = x¹ is a homomorphism for any a group (G, *). (f). For any two groups (G, *) and ( *), we have Gx G'G'xG. (g). The map : (C, +) → defined by p(x + y) = x +y is an epimorphism. (h). There are 5 subgroups of 42/64Z under the usual addition. (i). Let (Z, +) be the group of integers. The map 4 : Z× Z → Z defined by (a, b) = a - b is a homomorphism and ker(4) = {(a, a): a € Z}. (j). Z/nZ Zn, for any positive integer n (under addition). Dr bratam Alaluboot
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:8. Prove or disprove
(a). There is a homomorphism between any two groups.
(b). There is a finite group
to an infinite group.
(c). Any two finite groups the same order are isomorphic.
(d). There is an abelian group isomorphic to a non-abelian group.
(e). The map
G defined by p(x) = x¹ is a homomorphism
for any a group (G, *).
(f). For any two groups (G, *) and (
+), we have Gx G'G'XG.
(g). The map : (C, +) →
defined by (x + y) = x + y is
an epimorphism.
(h). There are 5 subgroups of 47/64Z under the usual addition.
(i). Let (Z, +) be the group of integers. The map 4 : Z× Z → Z
defined by (a, b) = a - b is a homomorphism and ker(4) =
{(a, a): a € Z}.
(j). Z/nZ Zn for any positive integer n (under addition).
Dr
bratam
Alaluboot
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