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{~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.

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

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.

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.

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.

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.

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