1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
LenKa [72]
3 years ago
15

Let H be a subgroup of a group G. We call H characteristic in G if for any automorphism σ∈Aut(G) of G, we have σ(H)=H.

Mathematics
1 answer:
choli [55]3 years ago
6 0

Answer:Problem 1. Let G be a group and let H, K be two subgroups of G. Dene the set HK = {hk : h ∈ H,k ∈ K}.

a) Prove that if both H and K are normal then H ∩ K is also a normal subgroup of G.

b) Prove that if H is normal then H ∩ K is a normal subgroup of K.

c) Prove that if H is normal then HK = KH and HK is a subgroup of G.

d) Prove that if both H and K are normal then HK is a normal subgroup of G.

e) What is HK when G = D16, H = {I,S}, K = {I,T2,T4,T6}? Can you give geometric description of HK?

Solution: a) We know that H ∩ K is a subgroup (Problem 3a) of homework 33). In order to prove that it is a normal subgroup let g ∈ G and h ∈ H ∩ K. Thus h ∈ H and h ∈ K. Since both H and K are normal, we have ghg−1 ∈ H and ghg−1 ∈ K. Consequently, ghg−1 ∈ H ∩ K, which proves that H ∩ K is a normal subgroup.

b) Suppose that H G. Let K ∈ k and h ∈ H ∩ K. Then khk−1 ∈ H (since H is normal in G) and khk−1 ∈ K (since both h and k are in K), so khk−1 ∈ H ∩ K. This proves that H ∩ K K.

c) Let x ∈ HK. Then x = hk for some h ∈ H and k ∈ K. Note that x = hk = k(k−1hk). Since k ∈ K and k−1hk ∈ H (here we use the assumption that H G), we see that x ∈ KH. This shows that HK ⊆ KH. To see the opposite inclusion, consider y ∈ KH, so y = kh for some h ∈ H and k ∈ K. Thus y = (khk−1)k ∈ HK, which proves that KH ⊆ HK and therefoere HK = KH. To prove that HK is a subgroup note that e = e · e ∈ HK. If a,b ∈ HK then a = hk and b = h1k1 for some h,h1 ∈ H and k,k1 ∈ K. Thus ab = hkh1k1. Since HK = KH and kh1 ∈ KH, we have kh1 = h2k2 for some k2 ∈ K, h2 ∈ H. Consequently,

ab = h(kh1)k1 = h(h2k2)k1 = (hh2)(k2k1) ∈ HK

(since hh2 ∈ H and k2k1 ∈ K). Thus HK is closed under multiplication. Finally,

Step-by-step explanation:

You might be interested in
A B C D which one is it ?
kirill115 [55]

Answer:

c.

Step-by-step explanation:


3 0
3 years ago
Read 2 more answers
Consider the function f(x) = -2x + 5 what is f(5)
zepelin [54]

Answer:

\displaystyle -5 = f(5)

Step-by-step explanation:

\displaystyle -2[5] + 5 = -10 + 5 = -5

I am joyous to assist you anytime.

4 0
3 years ago
Read 2 more answers
There are 120 people at a party 1/3 leave work out the number of people still at the party
Tpy6a [65]

Answer:

80

Step-by-step explanation:

1/3 of 120 is 40

If 40 leave

People left in party=120-40=80

7 0
3 years ago
Read 2 more answers
NEED HELP!!!!<br><br> Please
AnnZ [28]

Answer:

# 1,3 and 6

Step-by-step explanation:

4 0
2 years ago
Question 5 of 10
Alona [7]
I don’t understand?????
5 0
2 years ago
Other questions:
  • Z/m − z/n =1, if m≠n
    14·1 answer
  • Determine the solution set of x^2-80=0.
    15·2 answers
  • (I really need help with this, serious answers only please!)
    8·1 answer
  • If the P-value of a test of significance is greater than the level of significance, then which of these conclusions are appropri
    12·1 answer
  • Jake drove to California at a rate of 60 mph. If he stopped for gas 45 minutes into the trip, how far did he drive up to that po
    9·1 answer
  • I need help on this question
    7·2 answers
  • Elimination method<br> a-b=3 <br> 2a+3b=b
    9·1 answer
  • Y varies directly as x, if y=12 when x=5, find y when x=3
    9·1 answer
  • HELP ME AS FAST AS YOU CAN!
    14·2 answers
  • How do you solve 5(r+9)−2(1−r)=1<br> need quickly
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!