Suppose that H is a nonempty subset of a group G with the...

Suppose that $H$ is a nonempty subset of a group $G$ with the property that if $a$ and $b$ belong to $H$ then $a^{-1} b^{-1}$ belongs to $H .$ Prove or disprove that this is enough to guarantee that $H$ is a subgroup of $G$.

Key Concepts

Closure Properties

Closure under certain operations (such as inversion and the group product) is essential to the subgroup structure. The problem provides a specific closure property—that for any a, b in H, a?¹b?¹ is in H—which must be analyzed to see if it forces closure under the operations required for a subgroup.

Standard Subgroup Test

One common subgroup test states that a nonempty subset H of a group G is a subgroup if whenever a and b are in H, the element ab?¹ is also in H. This test encapsulates the need for closure under the group operation and the existence of inverses. In the given problem, a similar but different condition (closure under a?¹b?¹) is provided, and analyzing how this compares to the standard test is key.

Counterexample Construction

In abstract algebra, a powerful method to show that a condition is not sufficient is to construct a counterexample. Here one may consider a singleton subset {a} in a group where a is not the identity but satisfies a relation (for example, a³ = e) so that a?¹a?¹ equals a, hence fulfilling the property. Since a singleton that does not contain the identity cannot be a subgroup, such examples illustrate that the given property does not force H to be a subgroup.

Group

A group is an algebraic structure consisting of a set together with an operation that combines any two elements to form a third, and that satisfies three fundamental properties: associativity, the existence of an identity element, and the existence of inverses. This context matters here because all subgroup?considerations are based on these axioms.

Subgroup

A subgroup is a subset of a group that is itself a group under the same operation. To verify that a nonempty subset is a subgroup, one must check that it contains the identity, is closed under the group operation, and is closed under taking inverses. The problem asks whether a certain closure?like property is sufficient to guarantee that the subset is a subgroup.

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