Is the group of order 175 abelian?

In summary, to prove that a group of order 175 is abelian, we can use the Sylow theorems to show that the group has only one Sylow 2-subgroup and one Sylow 7-subgroup, both of which are abelian. Since the group is isomorphic to the direct product of these subgroups, it follows that the group itself must be abelian. This is because the direct product of abelian groups is always abelian, as commutators have trivial intersection.
  • #1
lmedin02
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Homework Statement


Prove that the group of order 175 is abelian.


Homework Equations





The Attempt at a Solution


|G|=175=527. Using the Sylow theorems it can be determined that G has only one Sylow 2-subgroup of order 25 called it H and only one Sylow 7-subgroups called it K. Thus, H and K are normal subgroups of G and G=H x K which is isomorphic to the direct product of H and K. Since |H|=52, then H is Abelian. Since K is of prime order then K is cyclic and therefore also Abelian.

I am not sure whether I can now conclude that G must be abelian since it is the external (or direct) product of abelian subgroups.
 
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  • #2
Can you try to prove that the direct product of abelian groups is abelian?
 
  • #3
direct products of abelian groups are abelian, this is obvious. Look at commutators and use the fact that they have trivial intersection.
 

1. What is the purpose of classifying finite groups?

The purpose of classifying finite groups is to organize and understand the different types of groups that exist. By classifying them, we can identify common properties and relationships between groups, which can aid in solving problems and making predictions in various areas of mathematics.

2. How are finite groups classified?

Finite groups are classified based on their structural properties, such as their order (number of elements), subgroups, and symmetry. Different classification methods include group tables, Cayley graphs, and group presentations.

3. What are the main types of finite groups?

The main types of finite groups are cyclic groups, abelian groups, and non-abelian groups. Cyclic groups have a single generator and follow a repeating pattern, abelian groups have commutative operations, and non-abelian groups have non-commutative operations.

4. How are finite groups represented?

Finite groups can be represented in various ways, such as through matrices, permutations, and group presentations. These representations can help us understand the structure and properties of a group and can also aid in calculations and proofs.

5. What is the significance of the classification of finite groups?

The classification of finite groups has many practical applications in mathematics, physics, and computer science. It helps identify patterns and symmetries in real-world systems, aids in the development of new algorithms and encryption methods, and has connections to other areas of mathematics such as number theory and geometry.

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