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mathusers

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Hey there, i have a question on the center of a group, regarding group theory.

QUESTION:

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The centre Z(G) of a group G is defined by [itex]Z(G) = g \epsilon G: \forall x \epsilon G, xg = gx[/itex]

(i) Show that Z(G) is normal subgroup of G

(ii) By considering the Class Equation of G acting on itself by conjugation show that if [itex] |G| = p^n[/itex] ( p prime) then [itex] Z(G) \neq {1} [/itex]

(iii) If G is non abelian show that G/Z(G) is not cyclic.

(iv) Decude that any group of order [itex] p^2[/itex] is abelian.

(V) Deduce that a gorup of oder [itex] p^2 [/itex] is isomorhpic either to [itex] C_{p^2}[/itex] or to [itex] C_p \times C_p[/itex]

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WHAT I HAVE SO FAR: PLEASE VERIFY THEM

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(i) is not too hard: we let [itex]x\in \text{Z}(G)[/itex] and so we prove that [itex]gxg^{-1} \in \text{Z}(G)[/itex] for any [itex]g\in G[/itex].

(ii) [itex] G \equiv |Z(G)| (mod p)[/itex] since Z(G) is a fixed point set.

Now [itex]|Z(G)| \equiv p^n(mod p)[/itex], |Z(G)|=0.

So Z(G) has atleast p elements.

(v) We let [itex]|G|=p^2[/itex]. We choose [itex]a\not = 1[/itex]. We form a subgroup [itex]H=\left< a \right>[/itex] if [itex]H = G[/itex]. This implies that the group is cyclic and so the proof is complete. If this is not the case then we pick [itex]b\in G\setminus H[/itex] and form [itex]K=\left< b\right>[/itex]. This means [itex]H\cap K = \{ 1\}[/itex] which further implies [itex]HK = G[/itex]. Also, since the group is abelian [itex]H,K\triangleleft G[/itex]. So [itex]G\simeq H\times K \simeq \mathbb{Z}_p \times \mathbb{Z}_p[/itex].

please verify these and help me out with the rest. very many thanks :)

QUESTION:

--------------------------------------

The centre Z(G) of a group G is defined by [itex]Z(G) = g \epsilon G: \forall x \epsilon G, xg = gx[/itex]

(i) Show that Z(G) is normal subgroup of G

(ii) By considering the Class Equation of G acting on itself by conjugation show that if [itex] |G| = p^n[/itex] ( p prime) then [itex] Z(G) \neq {1} [/itex]

(iii) If G is non abelian show that G/Z(G) is not cyclic.

(iv) Decude that any group of order [itex] p^2[/itex] is abelian.

(V) Deduce that a gorup of oder [itex] p^2 [/itex] is isomorhpic either to [itex] C_{p^2}[/itex] or to [itex] C_p \times C_p[/itex]

--------------------------------------

WHAT I HAVE SO FAR: PLEASE VERIFY THEM

--------------------------------------

(i) is not too hard: we let [itex]x\in \text{Z}(G)[/itex] and so we prove that [itex]gxg^{-1} \in \text{Z}(G)[/itex] for any [itex]g\in G[/itex].

(ii) [itex] G \equiv |Z(G)| (mod p)[/itex] since Z(G) is a fixed point set.

Now [itex]|Z(G)| \equiv p^n(mod p)[/itex], |Z(G)|=0.

So Z(G) has atleast p elements.

(v) We let [itex]|G|=p^2[/itex]. We choose [itex]a\not = 1[/itex]. We form a subgroup [itex]H=\left< a \right>[/itex] if [itex]H = G[/itex]. This implies that the group is cyclic and so the proof is complete. If this is not the case then we pick [itex]b\in G\setminus H[/itex] and form [itex]K=\left< b\right>[/itex]. This means [itex]H\cap K = \{ 1\}[/itex] which further implies [itex]HK = G[/itex]. Also, since the group is abelian [itex]H,K\triangleleft G[/itex]. So [itex]G\simeq H\times K \simeq \mathbb{Z}_p \times \mathbb{Z}_p[/itex].

please verify these and help me out with the rest. very many thanks :)

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