Understanding Subgroups in Group Theory

In abstract algebra, a group is a set equipped with a binary operation that satisfies four conditions: closure, associativity, identity element existence, and inverse element existence. A subset H of a group (G, ∗) is called a subgroup of G if H is itself a group under the same operation ∗.

For a non-empty subset H of a group G to be a subgroup, it must satisfy the following conditions:

• Closure Property: For all a, b ∈ H, a ∗ b ∈ H.
• Identity Element: The identity element of G must be in H.
• Inverse Element: For all a ∈ H, the inverse a^-1 in G must also be in H. This is closure under the inverse element operation.

Alternatively, there is a single, more concise test for a non-empty subset H to be a subgroup: for all a, b ∈ H, a ∗ b^-1 ∈ H. This single condition implies all three of the above (assuming H is non-empty).

Analyzing the Given Options for Subgroup Conditions

Let's look at the given options in the context of a subset H being a group (i.e., a subgroup):

1. a, b ∈ H ⇒ a ∗ b ∈ H: This is the closure property. This is a necessary condition for H to be a group, but it is not sufficient on its own. For example, the set of natural numbers {1, 2, 3, ...} is closed under addition, but it is not a group because it lacks an identity element (0) and inverse elements.
2. a ∈ H ⇒ a^-1 ∈ H: This condition states that for every element 'a' in H, its inverse element 'a^-1' (which exists in the larger group G) must also belong to H. This property, known as closure under inverses, is also a necessary condition for H to be a group. If H is a group, then by definition, every element in H must have an inverse *within* H. Since H is a subset of G, the inverse in H must be the same as the inverse in G. Thus, if H is a group, this condition must hold. While not sufficient alone (H also needs closure and identity), it's a fundamental requirement for the structure of a group.
3. a, b ∈ H ⇒ a ∗ b^-1 ∈ H: As mentioned earlier in group theory, this single condition, when combined with the fact that H is non-empty, is a standard test for H to be a subgroup. It implies both closure under the operation and the existence of identity and inverses within H. Among the choices, this is the strongest single condition that implies H is a group (assuming H is non-empty).
4. H contains the identity element: The presence of the identity element is necessary for H to be a group, but it is not sufficient. For example, the set {0, 1} is a subset of the integers under addition (&Z, +), and it contains the identity 0, but it's not a group (e.g., it's not closed: 1+1 = 2 ∉ {0, 1}).

Considering the options, Option 2 highlights a crucial necessary property for a subset H to be a group under the inherited operation – the existence of inverses within H for all its elements. While Option 3 is the most comprehensive single test for a subgroup, Option 2 explicitly states the requirement that every element in H must have its inverse element also residing in H, a property indispensable for H to function as a group itself. Therefore, if H is a group, this condition must be met.