An infinite group is a group with infinitely many elements. An element of infinite order proves that its group is infinite, but the converse need not hold: an infinite group can have all elements of finite order.
An infinite group whose every proper nontrivial subgroup is cyclic of a fixed prime order . Such groups exist for sufficiently large primes. This subgroup condition is stronger than the condition that every nonidentity element has order .
// Target: algebra.bigb
For a Tarski monster group, any nonidentity element generates a proper subgroup of order . Adjoining an element outside that subgroup generates the whole group. A proper nontrivial normal subgroup would have order ; conjugation into its finite automorphism group forces it to be central. Combining it with another order- cyclic subgroup would then give a proper subgroup of order , a contradiction.
// Target: algebra.bigb
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In group theory, a branch of abstract algebra, an **infinite group** is a group that contains an infinite number of elements. In other words, if the cardinality (size) of the group is not a finite number, then the group is classified as infinite. Infinite groups can be categorized into various types based on their structure and properties.