# Show that a group with no proper nontrivial subgroups is cyc

• Mr Davis 97
In summary, to show that a group with no proper nontrivial subgroups is cyclic, we can assume that the group has at least two elements and let any element besides the identity element generate a subgroup. Since the group has no proper nontrivial subgroups, this subgroup must be the entire group, making the group cyclic.
Mr Davis 97

## Homework Statement

Show that a group with no proper nontrivial subgroups is cyclic.

## The Attempt at a Solution

If a group G has no proper nontrivial subgroups, then its only subgroups are ##\{e \}## and ##G##. Assume that G has at least two elements, and let ##a## be any element besides ##e##. Then ##a## generates a subgroup of ##G##, but ##G## has no proper nontrivial subgroups, which means that ##a## must generate ##G##, so ##G## is cyclic.

I feel that I am on the right track, but I also don't feel like I am being rigorous enough.

Your proof is fine. I would just replace "##a## generates a subgroup of ##G##" by "##a## generates a non-trivial subgroup of ##G##", and observe that ##\langle a\rangle##, the subgroup generated by ##a##, cannot be proper, before stating that ##\langle a\rangle=G##.

Mr Davis 97

## What does it mean for a group to have no proper nontrivial subgroups?

When a group has no proper nontrivial subgroups, it means that the only subgroups of the group are the identity element and the group itself. In other words, there are no smaller groups contained within the larger group.

## How is this property related to the concept of a cyclic group?

A group with no proper nontrivial subgroups is always a cyclic group. This means that the group can be generated by a single element, and all other elements in the group are powers of this generator.

## What is the significance of this property in group theory?

This property is significant because it helps us classify and understand different types of groups. Groups with no proper nontrivial subgroups have unique properties and behave differently than other types of groups.

## Can you provide an example of a group with no proper nontrivial subgroups?

The simplest example of a group with no proper nontrivial subgroups is the cyclic group of order 2, denoted as C2. It only contains the identity element and itself, and is generated by any non-identity element within the group.

## How can you prove that a group has no proper nontrivial subgroups?

To prove that a group has no proper nontrivial subgroups, you can use the fact that any non-identity element in the group must generate the entire group. This means that there are no smaller, non-trivial subgroups contained within the group. Alternatively, you can also use the Lagrange's theorem, which states that the order of any subgroup must divide the order of the larger group. Since there are no proper nontrivial subgroups, the only possible order for a subgroup is 1, making it the identity element.

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