Question

Let G be a group (not necessarily an Abelian group) of order 425. Prove that G...

Let G be a group (not necessarily an Abelian group) of order 425. Prove that G must have an element of order 5. Note, Sylow Theorem is above us so we can't use it. We're up to Finite Orders. Thank you.

Homework Answers

Answer #1

Statement of Cauchy's theorem : Let G be a finite group and be a prime number . If divides the order of G then G has an element of order .

Here G is a group of order 425 , since 5 divides 245 and 5 is a prime so G has an elements of order 5 .

.

.

The proof of Cauchys theorem is not that difficult , if you want proof of the theorem please comment .

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
LetG be a group (not necessarily an Abelian group) of order 425. Prove that G must...
LetG be a group (not necessarily an Abelian group) of order 425. Prove that G must have an element of order 5.
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 group of order p^3. Prove that either G is abelian or its...
Let G be a group of order p^3. Prove that either G is abelian or its center has exactly p elements.
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 G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G...
Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G → G by φ(x) = x^ k . (a) Prove that φ is a group homomorphism. (b) Assume that G is finite and |G| is relatively prime to k. Prove that Ker φ = {e}.
Let p,q be prime numbers, not necessarily distinct. If a group G has order pq, prove...
Let p,q be prime numbers, not necessarily distinct. If a group G has order pq, prove that any proper subgroup (meaning a subgroup not equal to G itself) must be cyclic. Hint: what are the possible sizes of the subgroups?
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
: (a) Let p be a prime, and let G be a finite Abelian group. Show...
: (a) Let p be a prime, and let G be a finite Abelian group. Show that Gp = {x ∈ G | |x| is a power of p} is a subgroup of G. (For the identity, remember that 1 = p 0 is a power of p.) (b) Let p1, . . . , pn be pair-wise distinct primes, and let G be an Abelian group. Show that Gp1 , . . . , Gpn form direct sum in...
Prove that if G is a group with |G|≤5 then G is abelian.
Prove that if G is a group with |G|≤5 then G is abelian.
Suppose that G is abelian group of order 16, and in computing the orders of its...
Suppose that G is abelian group of order 16, and in computing the orders of its elements, you come across an element of order 8 and 2 elements of order 2. Explain why no further computations are needed to determine the isomorphism class of G. provide explanation please.