Question

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.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
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 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.
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 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 non-trivial finite group, and let H < G be a proper subgroup....
Let G be a non-trivial finite group, and let H < G be a proper subgroup. Let X be the set of conjugates of H, that is, X = {aHa^(−1) : a ∈ G}. Let G act on X by conjugation, i.e., g · (aHa^(−1) ) = (ga)H(ga)^(−1) . Prove that this action of G on X is transitive. Use the previous result to prove that G is not covered by the conjugates of H, i.e., G does not equal...
Let G be a finitely generated group, and let H be normal subgroup of G. Prove...
Let G be a finitely generated group, and let H be normal subgroup of G. Prove that G/H is finitely generated
Let G be an Abelian group and H a subgroup of G. Prove that G/H is...
Let G be an Abelian group and H a subgroup of G. Prove that G/H is Abelian.
Let H be a normal subgroup of G. Assume the quotient group G/H is abelian. Prove...
Let H be a normal subgroup of G. Assume the quotient group G/H is abelian. Prove that, for any two elements x, y ∈ G, we have x^ (-1) y ^(-1)xy ∈ H
Let G be a group, and H a subgroup of G, let a,b?G prove the statement...
Let G be a group, and H a subgroup of G, let a,b?G prove the statement or give a counterexample: If aH=bH, then Ha=Hb
Let G be a finite group. There are 2 ways of getting a subgroup of G,...
Let G be a finite group. There are 2 ways of getting a subgroup of G, which are {e} and G. Now, prove the following : If |G|>1 is not prime, then G has a subgroup other than the 2 groups which are mentioned in the above.