Groups : Isomorphism and Homomorphism

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

Show that a group G is simple if and only if every nontrivial group homomorphism G -> G1 is one-to-one.

© BrainMass Inc. brainmass.com October 16, 2018, 4:10 pm ad1c9bdddf
https://brainmass.com/math/linear-transformation/groups-isomorphism-and-homomorphism-15977

Solution Preview

Proof:
If G is simple, then G has no nontrivial normal subgroup. Then for any nontrivial group homomorphism f: G->G1, by the isomorphism theory, we have ...

Solution Summary

Group simplicity is investigated.

$2.19

Homomorphisms are examined.

EDIT: G(p) = {x in G : |x| = p^k for some k greater than or equal to 0}

** Please see the attachment for the complete problem description **

View Full Posting Details

Purchase Solution

$2.19

Related BrainMass Content

Some Problems Involving Homomorphisms of Abelian Groups

... set of all elements of each cyclic group whose order ... be isomorphic to H, since direct
products preserve isomorphism. ... Let G and H denote finite abelian groups. ...

Group Theory : Homomorphism, Subgroups, Abelian Groups and Group Order

... then H must be a normal subgroup of G. 7. Let G be an abelian group of order ... Write
down all direct products of cyclic groups that could be isomorphic to G ...

Isomorphism of a Group

... Notes:- (a) Isomorphism of a Group A homomorphism φ from G into G , that is
φ : G → G is said to be an isomorphism if φ is one-one, onto. ...

Groups

Give an example of groups H_i, K_j such that H_1xH_2 is isomorphic ... Let G be the additive
group Q of rational numbers ... then we have , but is not isomorphic to any ...

forgetful functor

... For example, consider Ab, the category of (additive) abelian groups. ... If Φ is a group
isomorsphism, then : must be a bijective map ... Then : is an isomorphism in ...

Group Theory : Homomorphisms, Kernels, Isomorphisms and Fields

... (c) Let G be the group of 2x2 ... d)For the two homomorphisms in (c), prove that the
two quotient groups G/N, where N is the corresponding kernel, are isomorphic. ...

finite Abelian group

... in the group, not the order of the group itself. ... 2. Homomorphism: Let G and H be
two groups. ... 4. Isomorphism: An isomorphism is a homomorphism that is bijective ...

Aut(G), the set of automorphisms of G, is also a group.

... Notes:- (a) Isomorphism of a Group A homomorphism from into , that is is said
to be an isomorphism if is one-one, onto. (b) Automorphism ...

Two questions in group theory

... The only groups of order 6 are and . ... Then So taking . Thus and the isomorphism
theorem gives us. ... 1) Let G be a finite group with n elements. ...

Kernel of Phi and Homomorphism

... the first isomorphism theorem: Z30/ker(phi) is isomorphic to G ... Since Z30 is a cyclic
group, and ker(phi) is ... by the fundamental theorem of cyclic groups: ker(phi ...