Group Actions: Prime Divisors & Smallest Prime | Dummit & Foote

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In summary, the text explains that if G is a finite group of order n, then any subgroup H of G whose index is p is normal. However, part of this proof is not obvious to me. All prime divisors (p-1)! are less than p, which follows from the fundamental theorem of arithmetic.
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Maths Lover
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hi ,

this result is from text , Abstract Algebra by Dummit and foote .
page 120

the result says , if G is a finite group of order n , p is the smallest prime dividing the order of G , then , any subgroup H of G whose index is p is normal

and the text gave the proof of this result , but a part of this proof is not obivous for me !

this part is ,all prime divisors (p-1)! are less than p .

why is this true ?!

can anyone explain please ?
 
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  • #2
all prime divisors (p-1)! are less than p

Do you mean that all prime divisors of (p-1)! are less than p? Are you familiar with the fundamental theorem of arithmetic? It follows immediately from what (p-1)! is that no prime larger than p can be a divisor. Write it out in full; is p in there anywhere?
 
  • #3
Number Nine said:
Do you mean that all prime divisors of (p-1)! are less than p? Are you familiar with the fundamental theorem of arithmetic? It follows immediately from what (p-1)! is that no prime larger than p can be a divisor. Write it out in full; is p in there anywhere?

yes , I'm familiar with it !
can you explain how does this follows from the fundamental theorem of arithmetic ?
 
  • #4
Maths Lover said:
yes , I'm familiar with it !
can you explain how does this follows from the fundamental theorem of arithmetic ?

It follows from the definition of the factorial. Write out the expansion of (p-1)!; p does not appear anywhere in the factorization. How could it? You're multiplying together a bunch of numbers less than p. None of them are going to multiply together to form p (it's prime).
 
  • #5
Number Nine said:
It follows from the definition of the factorial. Write out the expansion of (p-1)!; p does not appear anywhere in the factorization. How could it? You're multiplying together a bunch of numbers less than p. None of them are going to multiply together to form p (it's prime).

take this example !
6 can't divide 3,4,5
but 6 can divide 3*4*5= 60

p can't divide any factor but maybe it can do this with some products of them like the example above ! why not ??
 
  • #6
Maths Lover said:
take this example !
6 can't divide 3,4,5
but 6 can divide 3*4*5= 60

p can't divide any factor but maybe it can do this with some products of them like the example above ! why not ??

6 is not prime.
(p-1)! has a unique prime factorization. Write out the expansion of (p-1)!, as I said; p does not appear, nor is it a factor of any of the numbers that do appear. Again, review the fundamental theorem of arithmetic.
 
  • #7
Number Nine said:
6 is not prime.
(p-1)! has a unique prime factorization. Write out the expansion of (p-1)!, as I said; p does not appear, nor is it a factor of any of the numbers that do appear. Again, review the fundamental theorem of arithmetic.

yes , i can understand it now :) thanx
 

What are group actions and why are they important in mathematics?

Group actions are mathematical operations that map elements of a group to other elements within the same group. They are important because they provide a way to study the structure and properties of groups, which are fundamental mathematical objects that appear in many areas of mathematics and have numerous applications in science and engineering.

What is a prime divisor and how is it related to group actions?

A prime divisor is a prime number that divides another number without leaving a remainder. In group actions, prime divisors are used to classify groups and determine their properties. For example, the order of a group (the number of elements in the group) must be divisible by all of its prime divisors.

What is the smallest prime and why is it important in group actions?

The smallest prime is the prime number that comes after 2, which is 3. In group actions, the smallest prime is important because it is used to define a special type of group called a cyclic group, which has many important properties and applications in mathematics and other fields.

How are prime divisors and smallest prime related to each other in group actions?

In group actions, the prime divisors of a group's order determine the structure and properties of the group, while the smallest prime is used to define the cyclic subgroups of the group. The combination of these two concepts helps to classify groups and understand their behavior.

Can group actions be applied in other areas of science and engineering?

Yes, group actions have numerous applications in many areas of science and engineering, such as physics, chemistry, computer science, and cryptography. They can be used to study symmetry, patterns, and transformations in various systems and to solve problems in coding, data compression, and signal processing.

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