Question

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.

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

Homework Answers

Answer #1

Solution:

Given that:

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 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...
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 group with subgroups H and K. (a) Prove that H ∩ K...
Let G be a group with subgroups H and K. (a) Prove that H ∩ K must be a subgroup of G. (b) Give an example to show that H ∪ K is not necessarily a subgroup of G. Note: Your answer to part (a) should be a general proof that the set H ∩ K is closed under the operation of G, includes the identity element of G, and contains the inverse in G of each of its elements,...
f H and K are subgroups of a group G, let (H,K) be the subgroup of...
f H and K are subgroups of a group G, let (H,K) be the subgroup of G generated by the elements {hkh−1k−1∣h∈H, k∈K}. Show that : H◃G if and only if (H,G)<H
Suppose : phi :G -H is a group isomorphism . If N is a normal subgroup...
Suppose : phi :G -H is a group isomorphism . If N is a normal subgroup of G then phi(N) is a normal subgroup of H. Prove it is a subgroup and prove it is normal?
Prove that if A is a subgroup of G and B is a subgroup of H,...
Prove that if A is a subgroup of G and B is a subgroup of H, then the direct product A × B is a subgroup of G × H. Show all steps. Note that AXB is nonempty since the identity e is a part of A X B. Remains only to show that A X B is closed under multiplication and inverses.
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!
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.
(a) Prove or disprove: Let H and K be two normal subgroups of a group G....
(a) Prove or disprove: Let H and K be two normal subgroups of a group G. Then the subgroup H ∩ K is normal in G. (b) Prove or disprove: D4 is normal in S4.
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.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT