A characteristic subgroup of a normal subgroup is normal

Let G be a group and let H \leq K \leq G.

  1. Prove that if H is characteristic in K and K is normal in G, then H is normal in G.
  2. Prove that if H is characteristic in K and K is characteristic in G, then H is characteristic in G. Use this to prove that the Klein 4-group V_4 is characteristic in S_4.
  3. Give an example to show that if H is normal in K and K is characteristic in G then H need not be normal in G.

    1. Let \varphi_g denote the automorphism of G which conjugates by g. Since K is normal in G, we have \varphi_g[K] = gKg^{-1} = K; thus \varphi_g is an automorphism of K. Since H is characteristic in K, we thus have gHg^{-1} = \varphi_g[H] = H for all g \in G. Hence H is normal in G.
    2. Let \varphi be an automorphism of G. Since K is characteristic in G, we have \varphi[K] = K; thus \varphi is an automorphism of K. Since H is characteristic in K, we have \varphi[H] = H; thus H is characteristic in G.

      Now V_4 is the unique subgroup of order 4 in A_4, and A_4 is the unique subgroup of order 12 in S_4. Thus V_4 \leq A_4 and A_4 \leq S_4 are characteristic; hence V_4 is characteristic in S_4.

    3. Recall the subgroup lattice of A_4. We have V_4 \leq A_4 characteristic since V_4 is the unique subgroup of order 4, and Z_2 \leq V_4 is normal because V_4 is abelian. However, letting Z_2 = \langle (1\ 2)(3\ 4) \rangle and \varphi(\sigma) = (1\ 2\ 3) \sigma (1\ 3\ 2), note that \varphi[\langle (1\ 2)(3\ 4) \rangle] = \langle \varphi((1\ 2)(3\ 4)) \rangle = \langle (1\ 4)(2\ 3) \rangle. Thus Z_2 is not characteristic in A_4.
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