Solve Induction Problem: Show Group G of Order p^n is Cyclic

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In summary, the conversation discusses proving that a group G of order p^n, where p is prime, is cyclic if it has exactly one subgroup for each divisor of p^n. The solution involves using the G/Z(G) theorem to show that G/Z(G) is cyclic and then using the Fundamental Theorem of Finite Abelian Groups to show that G is isomorphic to Zp or Zp+Z(p^n-1). The conversation also clarifies the notation Z(G) as the center of the group G, which is non-trivial for finite p-groups. The inductive hypothesis is used on the quotient G/Z(G) to prove the claim.
  • #1
tyrannosaurus
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Homework Statement


Suppose the G is a group of order p^n, where p is prime, and G has exactly one subgroup for each divisor of p^n. Show that G is cyclic.


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The Attempt at a Solution


First, the inductive basis holds for n=1. Now for each k>=1, we assume that the claim is true for a group of order p^k where k < n. We shall show that the claim is also true for a group of order p^n.

Note that in a general case, the group of order p^n is not necessarily abelian. However, by the given inductive hypotheis, we show that the above group G of order p^n is abelian. We know that Z(G) is non-trivial for every nontrivial finite p-group and is a subgroup of G.

From this I am suppose to use the G/Z(G) theroem to show that G/Z(G) is cyclic and thus that G is abelian. However, I don't see how I this is true, for example what if |G|=p^3 and |Z(G)|=p, then G/Z(G)= p^2, which does not mean that G/Z(G) is cyclic. (If I remember right G/Z(G) has to be trivial cyclic)
From this I can use the Fundamental Theorem of Finite Abelian Groups to get the G is isomorphic to Zp+Z(p^n-1) or Zp and then conclude that it is isomorphic to Zp.
 
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  • #2
What are the divisors of [itex]p^{n}[/itex]?
 
  • #3
1, p, p to m where m divides n and p^n. We know that we can't have 1 because Z(G) needs more than one lement
 
  • #4
Every group has 2 subgroups, namely [itex]\{e\}[/itex] and [itex]G[/itex]. So, there is always a subgroup with 1 element.
 
  • #5
{e} is always a subgroup, but in are case Z(G) cannot equal e since are group G is of order p^n, where p is a prime.
 
  • #6
I'm sorry. Maybe it's trivial, but I am unfamiliar with the notation [itex]Z(G)[/itex]. What do you mean by it?
 
  • #7
Z(G) is the center of the group G, for a to be an element of Z(G) a has to commute with all elements of G. Z(G) can be trivial (i.e just e) in some cases, but in this one it can't since |G|=p^n where p is a prime (p0groups have non-trivial centers).
 
  • #8
You're probably supposed to use the inductive hypothesis on the quotient G/Z(G)
 

1. How do you prove that a group of order p^n is cyclic?

To prove that a group G of order p^n is cyclic, we need to show that G has an element of order p^n. This can be done using induction, where we first show that the statement is true for the base case of n=1. Then, assuming the statement is true for n=k, we can show that it is also true for n=k+1 by constructing an element of order p^(k+1) using the fact that G is a group of order p^k.

2. What is the significance of a group being cyclic?

A cyclic group is a group where every element can be generated by a single element, called a generator. This allows for simpler and more efficient calculations and proofs, as well as providing a deeper understanding of the group's structure.

3. Can a group of order p^n be cyclic for any prime p and natural number n?

Yes, a group of order p^n can be cyclic for any prime p and natural number n. This is because there exists a finite cyclic group of order n for every positive integer n.

4. How does induction help in solving the problem of showing a group of order p^n is cyclic?

Induction is a powerful proof technique that allows us to prove a statement for an infinite number of cases by only proving it for a few base cases. In this problem, induction allows us to reduce the proof of showing a group of order p^n is cyclic to a simpler proof for the base case of n=1, which can then be extended to all other cases using the inductive step.

5. Are there any other methods for proving that a group of order p^n is cyclic?

Yes, there are other methods for proving that a group of order p^n is cyclic, such as using the structure theorem for finite abelian groups. However, the use of induction is a common and efficient approach for this particular problem.

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