Can a group of order 98 have a subgroup of order 7?

  • Thread starter rayveldkamp
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In summary, Sylow's Theorems are a set of three theorems in group theory that help us count the number of subgroups and understand the structure of finite groups. They are significant because they allow us to break down complicated groups into smaller, more manageable subgroups. The three theorems have different focuses, but they all play a crucial role in determining the structure of a group. Sylow's Theorems are used in various mathematical fields and cannot be applied to infinite groups.
  • #1
rayveldkamp
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Hi
Why must a group of order 98 contain a subgroup of order 7?
I would think that Sylow's 1st theorem implies there exists at least one Sylow-7-subgroup of order 49 and at least one Sylow-2-subgroup of order 2 (since 98=2x7x7).
Thanks

Ray Veldkamp
 
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  • #2
7 is prime, 7 divides 98, hence by Cauchy's theorem there is an element of order 7. The subgroup generated by this element is of order 7.
 
  • #3
or because the stronger version of sylows theorems say there is always a subgroup of any prime power order that divides the order of the group, not just the maximal prime power order.
 

1. What are Sylow's Theorems?

Sylow's Theorems are a set of three theorems in group theory that provide a way to count the number of subgroups of a finite group and determine their structure.

2. What is the significance of Sylow's Theorems?

Sylow's Theorems are important because they provide a powerful tool for understanding the structure of finite groups. They allow us to break down a large, complicated group into smaller, more manageable subgroups.

3. What is the difference between the three Sylow's Theorems?

The first theorem states that for any prime number p, if pn divides the order of a group, then the group contains a subgroup of order pn. The second theorem states that all subgroups of a given order are conjugate to each other. The third theorem states that the number of subgroups of a given order is congruent to 1 mod p.

4. How are Sylow's Theorems used in group theory?

Sylow's Theorems are used to determine the structure of a finite group and to classify groups with certain properties. They are also used in the study of other mathematical fields such as number theory, geometry, and abstract algebra.

5. Can Sylow's Theorems be applied to infinite groups?

No, Sylow's Theorems only apply to finite groups. In fact, one of the conditions for the theorems to hold is that the group must be finite. There are other theorems and techniques that are used to study infinite groups.

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