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I suspect it is quite simple and works by contradiction, but I can't see why for a group homomorphism f, ker(f) = {e} ==> f is injective. Any help?
I hadn't realized that.matt grime said:why would you want ax=xa anyway? that is not the condition of normality of a subgroup. H is normal if for gx in gH there is an element y of H such that yg=gx, there is no requirement that x=y.
May I have a hint?matt grime said:did you redo the proof of the last problem to be a one line proof NOT by contradiction? it would be helpful to your understanding to do it.
Sure you have. The basic group operation is often called multiplication unless otherwise specified. But if the use of the word "multiplication" confuses you, ignore that word and replace that part of my post with:quasar987 said:Ooh, I see .
(I haven't seen multiplication yet AKG)
matt grime said:i think it's quite straight forward, if H is the inerse image of H', then gx in H...
matt grime said:...implies [tex]f(gx) \in f(gH)=f(g)H'=H'f(g)[/tex] which implies that there is a y in H such that f(yg)=f(xg)
matt grime said:...so that, pulling back by f gx is in Hg too.
If a is an element of a group G, there is always a homomorphism from Z to G which sends 1 to a. When is there a homomorphism from [itex]\mathbb{Z}_n[/itex] to G which sends [1] to a? What are the homomorphisms from [itex]\mathbb{Z}_2[/itex] to [itex]\mathbb{Z}_6[/itex]?
Suppose G is a group and g is an element of G, g [itex]\neq[/itex] e. Under what conditions on g is there a homomorphism f : Z_7 --> G with
f([1]) = g ?
A group homomorphism is important in mathematics because it allows us to study the structure and relationships between different groups. By understanding how group homomorphisms work, we can better understand the properties and behaviors of groups, which are fundamental objects in abstract algebra and other areas of mathematics.
A group homomorphism is a function between two groups that preserves the group structure. In other words, it maps elements from one group to elements in another group in a way that respects the group operations. This means that the result of combining two elements in the first group will be the same as combining their corresponding images in the second group.
A group homomorphism is a function that preserves the group structure, while an isomorphism is a bijective homomorphism. This means that an isomorphism not only preserves the group operations, but also preserves the group's identity and inverses. In other words, an isomorphism is a one-to-one mapping between two groups that also preserves their algebraic properties.
Yes, a group homomorphism can be surjective but not injective. This means that the function maps every element in the first group to an element in the second group, but there may be multiple elements in the first group that map to the same element in the second group. In other words, the function is "onto" but not "one-to-one".
A group homomorphism is closely related to other mathematical concepts such as group actions, group representations, and ring homomorphisms. Group actions are similar to group homomorphisms in that they also preserve the group structure, but they act on a set rather than mapping between groups. Group representations are a way of studying groups by representing their elements as matrices or linear transformations, and group homomorphisms can be used to compare and relate different group representations. Ring homomorphisms are similar to group homomorphisms, but they preserve the structure of rings instead of groups.