Question

let G be a group of order 18. x, y, and z are elements of G. if | < x, y >| = 9 and o(z) = order of z = 9, prove that G = < x, y, z >

Answer #1

8.4: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is surjective then g is surjective.
8.5: Let f : X → Y and g : Y→ Z be bijections. Prove that if
composition g o f is bijective then f is bijective.
8.6: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is bijective then f is...

Let G be a group. Define Z(G) ={x∈G|xg=gx for all g∈G}, that is
Z(G) is the set of elements commuting with all the elements of G.
We call Z(G) the center of G. (In German, the word for
center is Zentrum, hence the use of the “Z”.)
(a) Show that Z(G) is a subgroup of G.
(b) Show that Z(G) is an abelian group.

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).

Using field and order axioms prove the following theorems:
(i) Let x, y, and z be elements of R, the
a. If 0 < x, and y < z, then xy < xz
b. If x < 0 and y < z, then xz < xy
(ii) If x, y are elements of R and 0 < x < y, then 0 <
y ^ -1 < x ^ -1
(iii) If x,y are elements of R and x <...

Let G be a graph with x, y, z є V(G). Prove that if G contains
an x, y-path and a y, z-path, then it contains an x, z-path.

Let G = <a> be a cyclic group of order 12. Describe
explicitly all elements of Aut(G), the group of automorphisms of G.
Indicate how you know that these are elements of Aut(G) and that
these are the only elements of Aut(G).

Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A 4
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

Let X, Y ⊂ Z and x, y ∈ Z
Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )

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 a group of order p^3. Prove that either G is abelian or
its center has exactly p elements.

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