Question

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

Answer #1

Doubts are welcome.

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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.

The subgroup relation ≤ on the set of subgroups G is reflexive,
transitive, and anti-symmetric.

(a) Prove or disprove: Let H and K be two normal subgroups of a
group G. Then the subgroup H ∩ K is normal in G. (b) Prove or
disprove: D4 is normal in S4.

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

Disprove: The following relation R on set Q is either reflexive,
symmetric, or transitive. Let t and z be elements of Q. then t R z
if and only if t = (z+1) * n for some integer n.

(a) Prove or disprove: if H and K are subgroups of G, then H ∩ K
is a subgroup of G.
(b) Prove or disprove: if H is an abelian subgroup of G, then G
is abelian

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

a)
Let R be an equivalence relation defined on some set A. Prove
using induction that R^n is also an equivalence relation. Note: In
order to prove transitivity, you may use the fact that R is
transitive if and only if R^n⊆R for ever positive integer n
b)
Prove or disprove that a partial order cannot have a cycle.

Prove that if the relation R is symmetric, then its transitive
closure, t(R)=R*, is also symmetric. Please provide step by step
solutions

5. Prove or disprove the following statements:
(a) Let R be a relation on the set Z of integers such that xRy
if and only if xy ≥ 1. Then, R is irreflexive.
(b) Let R be a relation on the set Z of integers such that xRy
if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.
(c) Let R and S be reflexive relations on a set A. Then,...

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