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Math Help - isomorphism type question

  1. #1
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    isomorphism type question

    Suppose |G| = 16 and Z(G) contains an element of order 4. What is the isomorphism type of G/Z(G)? I am sure that you have to use the fundamental theorem of abelian groups and the G/Z(G) theorem but I am not sure how to apply these to the problem?
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  2. #2
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    Quote Originally Posted by aabsdr View Post
    Suppose |G| = 16 and Z(G) contains an element of order 4. What is the isomorphism type of G/Z(G)? I am sure that you have to use the fundamental theorem of abelian groups and the G/Z(G) theorem but I am not sure how to apply these to the problem?

    SO |Z(G)|\geq 4\Longrightarrow \left|G/Z(G)\right|\leq \frac{16}{4}=4.

    Now, it can't be |Z(G)|=8 since then \left|G/Z(G)\right|=2\Longrightarrow G/Z(G) is cyclic non-trivial, which is impossible (this itself is a nice problem!), so it must be

    either |Z(G)|=16\iff G/Z(G)=1\iff G is abelian, or else |Z(G)|=4\Longrightarrow G/Z(G)\cong \mathbb{Z}_2\times \mathbb{Z}_2 since, as mentioned above, this factor

    group cannot be cyclic non-trivial.

    Tonio
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