Let G and H be groups,and :GH a homomorphism. (a) Show that ker is a subset of G

ID: 3147556 • Letter: L

Question

Let G and H be groups,and :GH a homomorphism.

(a) Show that ker is a subset of G.

_______________________________________________________________

(b) Fill in the blanks with appropriate theorem or definition

Claim: ker/G.

Proof:

1. By_____,it suffices to show that for any gG, ker = g (ker) g1. So,let g_____.

2. First we show that (ker) g (ker) g1. Let xg (ker) g1.

(a) By_____,there exists k ker such that x = gkg1.

(b) By_____, (x) =(gkg1).

(d) By_____, (x) =(g)eH(g)1.

(e) By_____, (x) = eH.

(f) By denition of the kernel,_____.

(g) Since_____, g (ker) g1ker.

3. Now we show the converse; that is,_____. Let kker.

(a) Let x = g1kg. Notice that if xker, then we would have what we want,since in this case_____.

(b) In fact, xker. After all,_____.

4. By_____,ker = g (ker) g1.

(a) (blank 1) to show ker is a subset of G, by definition : ker = { gG | (g)=e }

hence by definition, the elements of ker are nothing but the elements of G.

hence, ker G.

(b) to show ker is a normal subgroup of G

1. by definition of normal subgroup , let gG

2. (a) by definition

(b) by the fact that is a well-defined map

(d) by the definition of kernel of , (k) = eH, the identity element in H.

(e) by the fact that eHcommutes with every element in H, we have , (g)eH(g)1 = eH (g) (g)1 = eH (gg1) = eH (eG) = eH eH = eH

(f) By denition of the kernel, xker

(g) Since, xg (ker) g1 and xker , g (ker) g1 ker.

3. ker g (ker) g1

(a) if xker, then, x = g1kg , i.e. k = gxg-1 g ker g-1 , since, xker

(b) In fact, xker. After all x = g1kg => (x) = (g1kg) = (g1) (k) (g) = (g1) eH (g) = ((g))-1 (g) eH = eH eH = eH

, hence, (x) = eH => xker

4. by definition of set equality

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.