Proving a group is infinite

• sairalouise

sairalouise

I'm trying to show that there is not one sentence (formula) that if a group satisfies this formula it is equivalent to the group being infinite. I can show this in a hap hazard way analogous to the same problem in the empty language , but how do you use the fact that the model is a group and there are arbitrarily large groups?

Let s be a sentence such that for all groups G, G models s iff G is infinite. Then a group G models ~s iff G is finite. So every finite group models ~s, and so

{~s} U {axioms of group theory}

has arbitrarily large finite models (since there are arbitrarily large finite groups). But a standard compactness argument yields that

{~s} U {axioms of group theory}

has an infinite model G which would be a group that models both s and ~s, contradiction.

What is meant by "proving a group is infinite"?

Proving a group is infinite means showing that the group contains an infinite number of elements, rather than a finite number.

How can you determine if a group is infinite?

There are several methods for proving that a group is infinite. One common method is to show that the group has an element with an infinite order, meaning that when the element is multiplied by itself a certain number of times, it never reaches the identity element.

What is the importance of proving that a group is infinite?

Proving that a group is infinite can have significant implications for the properties and behavior of the group. For example, infinite groups can have more complex structures and can exhibit more diverse patterns and behaviors compared to finite groups.

What are some common examples of infinite groups?

Some common examples of infinite groups include the group of integers under addition, the group of real numbers under addition, and the group of rational numbers under multiplication.

Are there any strategies or techniques for proving that a group is infinite?

Yes, there are several strategies and techniques that can be used to prove that a group is infinite. These may include using algebraic properties and identities, induction, and proof by contradiction.