Group is a union of proper subgroups iff. it is non-cyclic

In summary, a finite group is the union of proper subgroups if and only if the group is not cyclic. This is because if the group is a union of proper subgroups, then none of the subgroups can represent all the elements of the group, thus making it not cyclic. Conversely, if the group is not cyclic, then it is the union of all the subgroups generated by each element in the group.
  • #1
gummz
32
2

Homework Statement



Prove that a finite group is the union of proper subgroups if and only if the group is not cyclic.

Homework Equations



None

The Attempt at a Solution


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" => "
If the group, call it G, is a union of proper subgroups, then, for every subgroup, there is at least one element of G that is not in that particular subgroup. But then we know that none of the subgroups can represent all the elements of G. Therefore, G is not cyclic.

" <= "
If the group is not cyclic, then no element a in G generates G. That means that G is the union of all the <a> subgroups for all a in G.

Is this correct?
 
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  • #2
Looks fine to me.
 

1. What is a group?

A group is a mathematical structure consisting of a set of elements and a binary operation that follows certain rules, such as closure, associativity, identity, and inverse. Groups are commonly used in algebra, geometry, and other areas of mathematics.

2. What is a subgroup?

A subgroup is a subset of a group that itself forms a group when using the same binary operation as the original group. It contains the identity element of the original group and is closed under the same operation.

3. What does it mean for a group to be non-cyclic?

A group is non-cyclic if it cannot be generated by a single element, meaning that there is no single element that, when combined with itself repeatedly, can produce all the elements of the group. Non-cyclic groups have a more complex structure and cannot be easily represented by a single element.

4. How does a group being a union of proper subgroups relate to it being non-cyclic?

If a group is a union of proper subgroups, it means that the group can be broken down into smaller subgroups that are not equal to the whole group. This is a characteristic of non-cyclic groups, as cyclic groups can only be broken down into subgroups that are equal to the whole group.

5. Why is the statement "Group is a union of proper subgroups iff. it is non-cyclic" important?

This statement is important because it provides a useful criterion for determining whether a group is non-cyclic. Instead of directly trying to prove that a group is non-cyclic, we can show that it is a union of proper subgroups, which is typically easier to do. This statement also highlights the connection between the structure of a group and its cyclic properties.

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