Question

Let G be a group and define the center (of G) Z(G) = {a ∈ G | xa = ax, ∀ x ∈ G}

a. Prove that Z(G) forms a subgroup of G.

b. If G is abelian, show that Z(G) = G.

c. What is the center of S_{3}

Answer #1

Let G be a group and define the center Z9G) = {a ∈ G | xa = ax,
∀ x ∈ G}.
a. Prove that Z(G) forms a subgroup of G.
b. What is the center of Z7?

Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G →
G by φ(x) = x^ k . (a) Prove that φ is a group homomorphism. (b)
Assume that G is finite and |G| is relatively prime to k. Prove
that Ker φ = {e}.

Let G be an Abelian group and let H be a subgroup of G Define K
= { g∈ G | g3 ∈ H }. Prove that K is a subgroup of G
.

Let G be a group and let a ∈ G. The set CG(a) = {x ∈
G | xa = ax} of all elements that commute with a is called the
Centralizer of a in G.
(b) Compute CG(a) when G = S3and a = (1,
2).
(c) Compute CG(a) when G = S4 and a = (1,
2).
(d) Prove that Z(G) = ∩a∈GCG(a).

Let G be an abelian group, let H = {x in G | (x^3) = eg}, where
eg is the identity of G. Prove that H is a subgroup of G.

Let G be an Abelian group and H a subgroup of G. Prove that G/H
is Abelian.

let g be a group. let h be a subgroup of g. define
a~b. if ab^-1 is in h. prove ~ is an equivalence relation on g

: (a) Let p be a prime, and let G be a finite Abelian group.
Show that Gp = {x ∈ G | |x| is a power of p} is a subgroup of G.
(For the identity, remember that 1 = p 0 is a power of p.) (b) Let
p1, . . . , pn be pair-wise distinct primes, and let G be an
Abelian group. Show that Gp1 , . . . , Gpn form direct sum in...

Let H be a normal subgroup of G. Assume the quotient group G/H
is abelian. Prove that, for any two elements x, y ∈ G, we have x^
(-1) y ^(-1)xy ∈ H

Let G be a ﬁnite Abelian group and let n be a positive divisor
of|G|. Show that G has a subgroup of order n.

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