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Math Help - Homomorphisms

  1. #1
    Super Member Aryth's Avatar
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    Homomorphisms

    Let G_1 and G_2 be groups with identities e_1 and e_2, respectively. Let f: G_1 \to G_2 be a homomorphism of groups. Let K be the kernel of f. Let x \in K. Which of the following is/are necessarily true?

    a) x = e_1
    b) x = e_2
    c) f(x) = e_2
    d) f(x) = e_1
    e) f(x) = e_1e_2

    I've never taken abstract algebra before so I have never seen anything like this. Any help would be greatly appreciated.
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  2. #2
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    Re: Homomorphisms

    The Kernel takes an element from the first group and maps it to the identity in the second group.
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    Re: Homomorphisms

    by definition, ker(f) = \{g \in G_1: f(g) = e_2\}. so clearly (c) is always true.

    (a) is sometimes true, but not always. for example, let G_1 = \mathbb{Z} and G_2 = \mathbb{Z}_5. the map:

    f:\mathbb{Z} \to \mathbb{Z}_5 given by f(x) = x\ (\text{mod } 5) is a homomorphism, but e_1 = 0 and 5 \in ker(f),

    but clearly 5 \neq 0.

    (b) is NEVER true. the kernel of f lies entirely in G1 (unless G1 is a subgroup of G2, in which case e_1 = e_2).

    (d) is likewise NEVER true, f(x) is always in G2 (but see above).

    (e) if G1, G2 are distinct groups, e_1e_2 isn't even defined.
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    Re: Homomorphisms

    Quote Originally Posted by Deveno View Post
    (a) is sometimes true, but not always. for example, let G_1 = \mathbb{Z} and G_2 = \mathbb{Z}_5. the map:

    f:\mathbb{Z} \to \mathbb{Z}_5 given by f(x) = x\ (\text{mod } 5) is a homomorphism, but e_1 = 0 and 5 \in ker(f),

    but clearly 5 \neq 0.
    Actually, I'm pretty sure the identity in the first group is always mapped to the identity in the second group. I believe that's a property of group homomorphisms.
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  5. #5
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    Re: Homomorphisms

    Quote Originally Posted by JSB1917 View Post
    Actually, I'm pretty sure the identity in the first group is always mapped to the identity in the second group. I believe that's a property of group homomorphisms.
    yes, but an element of the kernel K is not necessarily the identity. x COULD be the identity ((a) COULD be true), but it is not NECESSARILY true.
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  6. #6
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    Re: Homomorphisms

    Quote Originally Posted by Deveno View Post
    yes, but an element of the kernel K is not necessarily the identity. x COULD be the identity ((a) COULD be true), but it is not NECESSARILY true.
    Yeah, I misread it. I thought x was the only thing in the kernel. But a) would only be true if there was only one element, otherwise, as you noted, it could be many other things.
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