Find all the group homomorphisms from $G_1$ into $G_2$.

Asked Dec 13 '19 at 11:43

Active Dec 13 '19 at 11:43

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Find all the group homomorphisms from $G_1$ into $G_2$ wher $G_1=\{e, a, a^2|a^3=e\}$ and $G_2=\{e,b,b^2, b^3|b^4=e\}$.

So I know that $G_1=\langle a\rangle$ and $G_2=\langle b \rangle$, also $o(a)=3$ and $o(b)=4$ so $(3,4)=1$ that means there exists only one homomorphism. Which one?

asked Dec 13 '19 at 11:43

Atstovas

It's always worth checking the trivial cases. – CyclotomicField Dec 13 '19 at 11:46
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There is one homeomorphism that exists between any two groups, and if you don't know which one it is yet, I suggest that once you do find it, you memorize it. It's an important group homomorphism. – Arthur Dec 13 '19 at 12:06
@Arthur how should I do it? – Atstovas Dec 13 '19 at 12:08
@Atstovas What is the image of $e$ under an arbitrary homomorphism – justadzr Dec 13 '19 at 12:11
@YourongZang i have no idea – Atstovas Dec 13 '19 at 12:13
Well, then we start there. What is a homomorphism? Do you know that? – Arthur Dec 13 '19 at 12:18
@Atstovas by following Arthur's hints, you will be able to get the answer easily. – justadzr Dec 13 '19 at 12:19
@Arthur a homomorphism is a map between two structures with same type – Atstovas Dec 13 '19 at 12:24
@Atstovas You gotta be more specific when dealing with special algebraic structures. – justadzr Dec 13 '19 at 12:29
@Atstovas https://www.wikiwand.com/en/Group_homomorphism – justadzr Dec 13 '19 at 12:31
Yes, that's what a homomorphsim is, in the abstract sense. However, we know we are dealing with groups, so you can be more specific about what a homomorphism is in this context. What is a group homomorphsim? What requirements do we place on it? – Arthur Dec 13 '19 at 12:38
H(ab)=h(a)h(b) – Atstovas Dec 13 '19 at 12:49
Cool. And this means that what happens to the identity element? – Arthur Dec 13 '19 at 12:51
@Arthur so it is h(e) to e – Atstovas Dec 13 '19 at 12:52
Yes, that's right. And with your two groups, in your particular example, what can be done with $a$? What options are allowed for $h(a)$? – Arthur Dec 13 '19 at 13:38
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Now I dont know. Maybe h(a) =e – Atstovas Dec 13 '19 at 13:41

Find all the group homomorphisms from $G_1$ into $G_2$.

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