Question

Prove thatAnis a normal subgroup of Sn. Prove both that it is a subgroup AND that it is normal

Answer #1

If N is a normal subgroup of G and H is any subgroup of G, prove
that NH is a subgroup of G.

Please use two methods to prove that the set of odd permutations
of Sn is not a subgroup of Sn.

Prove or disprove: The relation "is-a-normal-subgroup-of" is a
transitive relation.

Do the odd permutations of Sn form a subgroup of
Sn? If so, provide a proof. If not, provide a
counterexample with justification

If H is a normal subgroup of G and |H| = 2, prove that H is
contained in the center of G.

Show that if G is a subgroup of Sn and |G| is odd, then G is a
subgroup of An, given n≥2

Suppose H is a normal subgroup of G where both H and G/H are
solvable groups. Prove that G is then a solvable group as well.

Let C be a normal subgroup of the group A and let D be a normal
subgroup of the group B.
(a) Prove that C × D is a normal subgroup of A × B
(b) Prove that the map φ : A × B → (A/C) × (B/D) given by φ((m, n))
= (mC, nD) is a group homomorphism.
(c) Use the fundamental homomorphism theorem to prove that (A ×
B)/(C × D) ∼= (A/C) × (B/D)

Suppose : phi :G -H is a group isomorphism . If N is a normal
subgroup of G then phi(N) is a normal subgroup of H. Prove it is a
subgroup and prove it is normal?

Let N be a normal subgroup of G. Prove or disprove the following
assertion:
N and G/N have composition series ----> G has a composition
series.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 2 minutes ago

asked 13 minutes ago

asked 19 minutes ago

asked 23 minutes ago

asked 37 minutes ago

asked 37 minutes ago

asked 37 minutes ago

asked 44 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago