Question

Show that if H is a subgroup of index 2 in a finite group G, then...

Show that if H is a subgroup of index 2 in a finite group G, then every left coset of H is also a right coset of H.

*** I have the answer but I am really looking for a thorough explanation. Thanks!

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let G be a finite group and H a subgroup of G. Let a be an...
Let G be a finite group and H a subgroup of G. Let a be an element of G and aH = {ah : h is an element of H} be a left coset of H. If B is an element of G as well show that aH and bH contain the same number of elements in G.
Let G be a finite group, and suppose that H is normal subgroup of G. Show...
Let G be a finite group, and suppose that H is normal subgroup of G. Show that, for every g ∈ G, the order of gH in G/H must divide the order of g in G. What is the order of the coset [4]42 + 〈[6]42〉 in Z42/〈[6]42〉? Find an example to show that the order of gH in G/H does not always determine the order of g in G. That is, find an example of a group G, and...
Let G be a finite group and H be a subgroup of G. Prove that if...
Let G be a finite group and H be a subgroup of G. Prove that if H is only subgroup of G of size |H|, then H is normal in G.
Show that if G is a group, H a subgroup of G with |H| = n,...
Show that if G is a group, H a subgroup of G with |H| = n, and H is the only subgroup of G of order n, then H is a normal subgroup of G. Hint: Show that aHa-1 is a subgroup of G and is isomorphic to H for every a ∈ G.
Let G be a finite group and let H be a subgroup of order n. Suppose...
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!
Problem 8. Suppose that H has index 2 in G. Prove that H is normal in...
Problem 8. Suppose that H has index 2 in G. Prove that H is normal in G. (Hint: Usually to prove that a subgroup is normal, the conjugation criterion (Theorem 17.4) is easier to use than the definition, but this problem is a rare exception. Since H has index 2 in G, there are only two left cosets, one of which is H itself – use this to describe the other coset. Then do the same for right 1 cosets....
(b) If H is a p-subgroup of a finite group G, prove that H is contained...
(b) If H is a p-subgroup of a finite group G, prove that H is contained in a Sylow p-subgroup of G. [Hint: Consider the H-conjugacy class equation for the set of all Sylowp-subgroups of G.]
a) Let H be a subgroup of a group G satisfying [G ∶ H] = 2....
a) Let H be a subgroup of a group G satisfying [G ∶ H] = 2. If there are elements a, b ∈ G such that ab ∈/ H, then prove that either a ∈ H or b ∈ H. (b) List the left and right cosets of H = {(1), (23)} in S3. Are they the same collection?
A subgroup H of a group G is called a normal subgroup if gH=Hg for all...
A subgroup H of a group G is called a normal subgroup if gH=Hg for all g ∈ G. Every Group contains at least two normal subgroups: the subgroup consisting of the identity element only {e}; and the entire group G. If G=S(n) show that A(n) (the subgroup of even permuations) is also a normal subgroup of G.
Let G be a finite group and let P be a Sylow p-subgroup of G. Suppose...
Let G be a finite group and let P be a Sylow p-subgroup of G. Suppose H is a normal subgroup of G. Prove that HP/H is a Sylow p-subgroup of G/H and that H ∩ P is a Sylow p-subgroup of H. Hint: Use the Second Isomorphism theorem.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT