Question

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 $G$.

Name: 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.
Uploaded: 2022-06-03T03:13:56-08:00
Duration: 1 min 26 s
Channel: Nick Johnson
Description: 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.

          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 $G$.

Added by Sara W.

Introductory and Intermediate Algebra for College Students 4th

Robert Blitzer 4th Edition

Instant Answer

Solved by Expert Nick Johnson

06/03/2022

Step 1

Step 1: To prove that $H$ is a subgroup of $G$, we need to show that it satisfies three criteria: closure under the group operation, the existence of the identity element in $H$, and the existence of inverses for every element in $H$. Show more…

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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 $G$.