Question

Let G be a group, and let a be in G. If |a| = 5, prove that the
centralizer of a is the centralizer of a^{3}

Answer #1

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 H be a subgroup of G, and N be the normalizer of H in G and
C be the centralizer of H in G. Prove that C is normal in N and the
group N/C is isomorphic to a subgroup of Aut(H).

Let G be a group (not necessarily an Abelian group) of order
425. Prove that G must have an element of order 5. Note, Sylow
Theorem is above us so we can't use it. We're up to Finite Orders.
Thank you.

Let G be a group, and H a subgroup of G, let a,b?G prove the
statement or give a counterexample:
If aH=bH, then Ha=Hb

Let G be a finitely generated group, and let H be normal
subgroup of G. Prove that G/H is finitely generated

Let G be a group and D = {(x, x) | x ∈ G}.
(a) Prove D is a subgroup of G.
(b) Prove D ∼= G. (D is isomorphic to G)

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

Let G be a finite group and H be a subgroup of G. Prove that if
H is
only subgroup of G of size |H|, then H is normal in G.

Let G be a group of order 4. Prove that either G is generated by
a single element or g^2 =1 for all g∈G.

Let G be an infinite simple p-group. Prove that Z(G) = 1.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 4 minutes ago

asked 4 minutes ago

asked 5 minutes ago

asked 13 minutes ago

asked 16 minutes ago

asked 18 minutes ago

asked 18 minutes ago

asked 20 minutes ago

asked 22 minutes ago

asked 26 minutes ago

asked 46 minutes ago

asked 51 minutes ago