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Math Help - Sylow subgroup of group of order 56

  1. #1
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    Sylow subgroup of group of order 56

    Let G be a group of order 56, T \in \text{Syl}_2(G), S \in \text{Syl}_7(G) be given. Then |T| = 8, |S| = 7. Assume S is not normal, so then T \lhd G. Show that T \cong C_2^3.

    Here's what I have so far. My idea is to show that each element of T has order dividing 2. Let z \in S be given such that \langle z \rangle = S. Then G / T = \{T z^i : i=0,\dots,6\} and \forall x \in T, (Tz)^x = Tz. I've also got that G = T \rtimes S, although I fail to seen how any of this is helping.

    Any advice?
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  2. #2
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    Quote Originally Posted by Giraffro View Post
    Let G be a group of order 56, T \in \text{Syl}_2(G), S \in \text{Syl}_7(G) be given. Then |T| = 8, |S| = 7. Assume S is not normal, so then T \lhd G. Show that T \cong C_2^3.

    Here's what I have so far. My idea is to show that each element of T has order dividing 2. Let z \in S be given such that \langle z \rangle = S. Then G / T = \{T z^i : i=0,\dots,6\} and \forall x \in T, (Tz)^x = Tz. I've also got that G = T \rtimes S, although I fail to seen how any of this is helping.

    Any advice?

    So you have that T\rtimes S\Longrightarrow there exists a non-trivial homomorphism S\rightarrow Aut(T) . Well, what are the possible orders of automorphisms groups of groups of order 8?

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    So you have that T\rtimes S\Longrightarrow there exists a non-trivial homomorphism S\rightarrow Aut(T) . Well, what are the possible orders of automorphisms groups of groups of order 8?

    Tonio
    So \exists \phi : S \to \text{Aut}(T) a group homomorphism such that \forall x, y \in T, \forall i, j \in \mathbb{Z}, x z^i y z^j = x \phi(z^i)(y) z^{i+j}. If \phi(z) = \text{id}_T then \forall x \in T, z^x = x^{-1} z x = x^{-1} \phi(z)(x) z = z, so as S = \langle z \rangle, S \lhd G, which is a contradiction. Therefore \phi(z) \ne \text{id}_T and so as o(\phi(z)) \ne o(z), o(\phi(z)) = 7. So it follows that \exists x \in T such that x,\phi(z)(x),\dots,\phi(z^6)(x) are distinct and have the same order. So T has at least 7 elements of a specific order.

    <br />
\begin{array}{c|ccc}<br />
& \text{Order} \, 2 & \text{Order} \, 4 & \text{Order} \, 8 \\<br />
\hline<br />
C_8 & 1 & 2 & 4 \\<br />
C_4 \times C_2 & 3 & 4 & 0 \\<br />
C_2^3 & 7 & 0 & 0 \\<br />
D_8 & 5 & 2 & 0 \\<br />
Q & 1 & 6 & 0<br />
\end{array}

    Therefore T \cong C_2^3.

    Thanks for the help!
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