Question

Suppose that a cyclic group G has exactly three subgroups: G itself, e, and a subgroup...

Suppose that a cyclic group G has exactly three subgroups: G itself, e, and a subgroup of order p, where p is a prime greater than 2. Determine |G|

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 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?
A group G is a simple group if the only normal subgroups of G are G...
A group G is a simple group if the only normal subgroups of G are G itself and {e}. In other words, G is simple if G has no non-trivial proper normal subgroups. Algebraists have proven (using more advanced techniques than ones we’ve discussed) that An is a simple group for n ≥ 5. Using this fact, prove that for n ≥ 5, An has no subgroup of order n!/4 . (This generalizes HW5,#3 as well as our counterexample from...
12.29 Let p be a prime. Show that a cyclic group of order p has exactly...
12.29 Let p be a prime. Show that a cyclic group of order p has exactly p−1 automorphisms
3. Suppose G = <a> is a cyclic group of order 15 (i.e. a has order...
3. Suppose G = <a> is a cyclic group of order 15 (i.e. a has order 15), and consider the subgroup K = <a^3>. (a) Determine the order of K. (b) Use Lagrange’s theorem to determine the index of K in G, and then list the distinct cosets of K in G explicitly. (Hint: Consider the cosets Ke and Kb for b does not ∈K
Textbook: Algebra A Graduate Course - Isaacs Prove that the group G has exactly two subgroups...
Textbook: Algebra A Graduate Course - Isaacs Prove that the group G has exactly two subgroups iff |G| is prime (and hence finite).
Suppose that H is a proper subgroup of G of index n, and that G is...
Suppose that H is a proper subgroup of G of index n, and that G is a simple group, that is, G has no normal subgroups except G itself and {1}. Show thatG can be embedded in Sn.
Suppose that G is a cyclic group, with generator a. Prove that if H is a...
Suppose that G is a cyclic group, with generator a. Prove that if H is a subgroup of G then H is cyclic.
1(a) Suppose G is a group with p + 1 elements of order p , where...
1(a) Suppose G is a group with p + 1 elements of order p , where p is prime. Prove that G is not cyclic. (b) Suppose G is a group with order p, where p is prime. Prove that the order of every non-identity element in G is p.
Let G be a group and let H1 and H2 be subgroups of G. Suppose that...
Let G be a group and let H1 and H2 be subgroups of G. Suppose that G is finite and H1 and H2 have orders p and q, respectively, where p and q are distinct primes. Prove that H1 ∩ H2 = {e}.
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.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT