Download Group Theory , normal subgroups and more Lecture notes Mathematics in PDF only on Docsity!
INTRODUCTION TO GROUPS AND SYMMETRY (MTH 203)
Lecture-14, (03-09-2019)
Normal Subgroup.
Definition 0.1. A subgroup N of a group G is called a normal subgroup of G if for every g โ G and n โ N , gngโ^1 โ N.
Lemma 0.2. Let G be a group and N be a subgroup of G. These following statements are equivalent.
(1) N is normal subgroup of G.
(2) For any a โ G, aN = N a.
Proof..
- 1 โ 7. Let a โ G and an โ aN , since anaโ^1 โ N implies that an โ N a. So we get aN โ N a, similarly we can prove N a โ aN. Hence aN = N a.
- 7 โ 1. Let aN = N a, for a โ G. Let g โ G, h โ N. Claim: To show H is normal, it is enough to show that ghgโ^1 โ N. Note that gh โ gN = N g, so we can write gh = kg โ N g for some k โ N. Hence ghgโ^1 = k โ H.
Problem-14.1: Let G be a group. Show that if n โ Z+^ and H is the unique subgroup of G of order n, then H is a normal subgroup of G.
Problem-14.2: Let G be a group and H be a subgroup G of index two. Show that H is normal subgroup of G.
Answer-1: Let G has two distinct coset H and aH, where a โ G and a /โ H (why ?). Let g โ G and h โ H.
- Case-1: If g โ H then ghgโ^1 โ H, (since H is subgroup).
- Case-2: Suppose g โ aH. Then there exists some k โ H such that g = ak. Now consider ghgโ^1 = akhkโ^1 aโ^1. Since khkโ^1 โ H say khkโ^1 = n โ H. (Just to make calculation easy) Claim: ghgโ^1 โ H. 1
2 MTH 203
Supoose ghgโ^1 โ/ H. So we have ghgโ^1 โ aH. Then there exists some l โ H such that akhkโ^1 aโ^1 = anaโ^1 = al. This implies that aโ^1 and hence a โ H. A contradiction of our assumtion that ghgโ^1 โ H. Now for g โ G and k โ H, we have shown that ghgโ^1 โ H, which implies that H is normal (by definition of normal subgroup).
Answer-2: Let x โ G.
- Case-1: x โ H. Then xH = H = Hx.
- Case-2: x 6 โ H. Then xH 6 = H and so xH = G โ H. Likewise Hx 6 = H so Hx = G โ H. Therefore, xH = G โ H = Hx. Thus the left and right of cosets match so H is normal in G. Use above Lemma. Also try without using Lemma. (I have discussed both way in class)).
Note: We will use problem 14. 2 as a fact many times and also give another proof later.
Remark 0.3. Note H = {(1, 2 , 3), (2, 1 , 3))} is a subgroup of index 3 in S 3. But H is not normal.
Can you show that If p is the smallest prime dividing the order of the group G and H is a subgroup of index p then H is normal?
Isomorphisms of groups. Let G 1 and G 2 are two groups and f : G 1 โ G 2 be a group homomorphism.
Definition 0.4. Define Ker(f ) = {g โ G 1 : f (g) = e 2 },
where e 2 is the identity of G 2.
Remark 0.5. Ker(f ) is a normal subgroup of G 1.
Definition 0.6. Define
Im(f ) = {f (g) : g โ G 1 } = {k โ G 2 : k = f (g), for some g โ G 2 }.
Remark 0.7. Im(f ) is a subgroup of G 2.
Problem-14.2: Is Im(f ) a normal subgroup of G 2?
Definition 0.8. A group homomorphism f : G 1 โ G 2 is called an isomorphism if f is one-one and onto.
Excercise-14.1: Find all group isomorphism from (Z, +) to (Z, +).
Excercise-14.2: Show that f : (Z, +) โ (Z, +), n 7 โ 3 n is not group isomorphism.
