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Let $G$ be a group. A subgroup $H$ of $G$ is called characteristic if $\phi(H ) ⊂ H$ for all automorphisms $ϕ$ of $G$. Pick out the true statement(s):
(a) Every characteristic subgroup is normal.
(b) Every normal subgroup is characteristic.
(c) If $N$ is a normal subgroup of a group $G$, and $M$ is a characteristic sub-group of $N$, then $M$ is a normal subgroup of $G$.

I am completely stuck on it. Can somebody help me to solve the problem?

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Please use the homework tag when posting homework. –  Erick Wong Dec 31 '12 at 6:57
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2 Answers 2

1) A subgroup is normal if its fixed by conjugation. Conjugation by any element induces an automorphism, so?

2) Can you find a normal subgroup not fixed by an outer automorphism. Keep it simple look at $\mathbb Z_2 \times \mathbb Z_2$.

3) Now $N$ is fixed under conjugation by $G$ and conjugating on $N$ induces an outer automorphism which then fixes $M$.

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Theorem: Let $G=H\times K$, then $H\times 1$ and $1\times K$ are normal subgroups of $G$ but not necessarily characteristics in $G$.

Note that by taking $\alpha:G\to G, \alpha(x,y)=(y,x)$, it is an isomorphism and for $x\neq 1\in H$ $(x,1)^{\alpha}=(1,x)$ which is not in $H\times 1$. So 2 is not correct in any conditions. 1 and 3? Look at to another answer.

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