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Exercise :

(A.1) Let G be a group. The subgroup $Z(G) = \{z \in G | zg =gz \space \forall \space g \in G\}$ is called the center of $G$. Show that $Z(G)$ is a proper subgroup of $G$.

(A.2) Find the center of the dihedral group $D_3$.

(A.3) Show that the groups $D_3$ and $S_3$ are isomorphic. Using (A.2) find the center of $S_3$.

(A.4) Show that for all $n>2$ the center $Z(S_n)$ of the permutation group $S_n$ contains only the identity permutation.

Attempt :

(A.1)

From the definition of the identity element : $eg = ge = g \space \forall \space g \in G$. This means that $e \in Z(G) \Rightarrow Z(G) \neq \emptyset$.

Now, let $a,b \in Z(G)$. Then :

$$\forall \space g \in G: (ab)g = a(bg) = a(gb) = (ag)b = (ga)b = g(ab)$$

which means that $ab \in Z(G)$.

Finally, let $c \in Z(G)$. Then :

$\forall \space g \in G : cg=gc \Rightarrow c^{-1}(gc)c^{-1}=c^{-1}(gc)c^{-1} \Rightarrow egc^{-1} =c^{-1}ge \Rightarrow gc^{-1} = c^{-1}g \Rightarrow c^{-1} \in Z(G) $

which finally leads to $Z(G) \leq G$.

(A.2)

I found an elaboration for the center of the dihedral group $D_{2n}$ here but it won't work to find out the center of $D_3$, so I would appreciate any tips or links.

For (A.3)-(A.4) I am at loss on how to even start, so I would really appreciate any thorough answer or links with similar exercises - tips.

Sorry for not being able to provide an attempt but currently I'm going over problems that were in exams the previous years (for my semester exams) and it seems there are a lot of stuff that I have difficulty handling, which seem more weird that what we had handled this far.

Community
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Rebellos
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    For A.3, consider $D_6$ ($D_3$ in your notation) to be acting on the three vertices of an equilateral triangle. This induces an isomorphism. – Yuxiao Xie Jan 21 '18 at 15:19
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    By direct inspection, the center of $D_3$ is trivial. For a reference, K. Conrad, dihedral groups. See also this duplicate, and apply it for $n$ odd, $n=3$. For $Z(S_n)$ see this duplicate. – Dietrich Burde Jan 21 '18 at 15:21
  • For A.4, the following observation is useful. Let $\varphi\in S_n$. Then for all $(i\ j\ \cdots k)\in S_n$ we have $\varphi(i\ j\ \cdots k)\varphi^{-1}=(\varphi(i)\ \varphi(j) \cdots \varphi(k))$. – Yuxiao Xie Jan 21 '18 at 15:24

2 Answers2

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For A.2 your link will work, there's just a difference in notation - your $D_3$ is the link's $D_6$.

For A.3, Colescu's comment is a good start. The symmetries of an equilateral triangle can be reached via rotations and reflections ($D_3$) or permuting the vertices ($S_3$). Show that these give the same group of symmetries.

For A.4, consider an element $\phi \in S_n$ (as a permutation acting in a set of letters) which is not the identity. So we know there is at least one letter - call it $a$ - which is not fixed. Say $\phi (a)=b$. There are two cases - $\phi(b)=a$ and $\phi(b)=c \neq a$. Let $\xi$ be the permutation which switches $a$ and $b$ (and leaves all else fixed) and $\psi$ the permutation which rotates $a,b,c$ cyclically. Verify that in the first case, $\psi\circ\phi(a)\neq\phi\circ\psi(a)$, and in the second case, $\xi\circ\phi(a)\neq\phi\circ\xi(a)$.

BallBoy
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A2:Use the relations $σ^3=e,τ^2=ε,στσ=τ^{-1}$.

A4: Do you know what conjugate ellements is and cycle representation? If yes then this is a very easy excersise sincethe center $Z(G)$ consists of precisly the ellements only conjugate to themselves. So if there is an element $σ$ in $S_n$ then this means that it has no conjugate. Can you continue from here?

Nick A.
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