Question

Please give examples for

(a) a non-commutative ring

(b) a subgroup H of a group G which is not normal in G

(c) a commutative ring R in which the zero ideal {0} is not
prime

(d) a nonzero proper ideal J in a commutative ring IZ which is not maximal in R

Answer #1

Let R be a commutative ring and let a ε R be a non-zero element.
Show that Ia ={x ε R such that ax=0} is an ideal of R. Show that if
R is a domain then Ia is a prime ideal

Let G be a non-trivial finite group, and let H < G be a
proper subgroup. Let X be the set of conjugates of H, that is, X =
{aHa^(−1) : a ∈ G}. Let G act on X by conjugation, i.e., g ·
(aHa^(−1) ) = (ga)H(ga)^(−1) .
Prove that this action of G on X is transitive.
Use the previous result to prove that G is not covered by the
conjugates of H, i.e., G does not equal...

(a) Show that H =<(1234)> is a normal subgroup of G=S4
(b) Is the quotient group G/H abelian? Justify?

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 finite group and let H be a subgroup of order n.
Suppose that H is the only subgroup of order n. Show that H is
normal in G.
Hint: Consider the subgroup aHa-1 of G.
Please explain in detail!

Let H be a subgroup of the group G. Deﬁne a set B by B = {x ∈ G
| xax−1 ∈ H for all a ∈ H}. Show that H < B.

In each part below, a group G and a subgroup H are given.
Determine whether H is normal in G. If it is, list the elements of
the quotient group G/H. (a) G = Z-15 × Z-20 and H = <(10,
17)> (b) G = S-6 and H = A-6 (c) G = S-5 and H = A-4

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

Let G be a group with subgroups H and K.
(a) Prove that H ∩ K must be a subgroup of G.
(b) Give an example to show that H ∪ K is not necessarily a
subgroup of G.
Note: Your answer to part (a) should be a general proof that the
set H ∩ K is closed under the operation of G, includes the identity
element of G, and contains the inverse in G of each of its
elements,...

Answer b please...
Let R be a ring and let Z(R) := {z ∈ R : zr = rz for all r ∈
R}.
(a) Show that Z(R) ≤ R. It is called the centre of R.
(b) Let R be the quaternions H = {a+bi+cj+dk : a,b,c,d ∈ R} and
let S = {a + bi ∈ H}. Show that S is a commutative subring of H,
but there are elements in H that do not commute with elements...

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