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To determine whether a set is a subgroup of a given group, it must satisfy certain criteria. These are known as the subgroup criteria. Specifically, for a subset \( H \) of a group \( G \), \( H \) must have:

• An identity element that is the same as in \( G \).
• Closure under the group operation.
• An inverse for every element in \( H \).

These conditions ensure that \( H \) behaves like a group on its own under the operation defined by \( G \). If all these conditions are fulfilled, \( H \) can be considered a subgroup of \( G \). It essentially means that within the larger group \( G \), the subset \( H \) has all the necessary properties to form a smaller group by itself. This forms the basis of subgroup detection in group theory.

In group theory, the closure property is one of the essential conditions that a subset must satisfy to be considered a subgroup. The closure property ensures that performing the group operation (like multiplication or addition) on any two elements within the subset results in another element of the same subset.For example, if we have a group \( G = \langle\mathbb{R}^*, \cdot\rangle\) and a subset \( H = \{2^n 3^m : n, m \in \mathbb{Z}\} \), checking closure involves confirming that for any \( a = 2^{n_1} 3^{m_1} \) and \( b = 2^{n_2} 3^{m_2} \) in \( H \), their product \( ab = 2^{n_1+n_2} 3^{m_1+m_2} \) is also in \( H \).Since the exponents \( n_1+n_2 \) and \( m_1+m_2 \) are integers (closed under addition), the closure property is satisfied. Thus, the operation never takes us out of the set \( H \), keeping the subset consistent with group behavior.

An identity element in the context of group theory is an element that, when used in a group operation with any element of the group, leaves that element unchanged. This identity element is crucial for a subset to qualify as a subgroup.In the group \( G = \langle \mathbb{R}^*, \cdot \rangle \), the identity element is 1, because multiplying any number by 1 does not change the number. To check if a subset \( H \) contains this identity, we need to verify that 1 can be expressed in terms of the subset's notation.For \( H = \{ 2^n 3^m : n, m \in \mathbb{Z} \} \), setting \( n = 0 \) and \( m = 0 \) gives \( 2^0 3^0 = 1 \). Since 1 is present in \( H \), it meets the identity element criterion, allowing us to further validate \( H \) as potentially being a subgroup of \( G \).

Having an inverse element for each element in a group is a vital property that allows the group to maintain its structure. In simple terms, for every element \( a \) in the group, there should be an element \( a^{-1} \) such that the product \( a \cdot a^{-1} = 1 \).To check for inverses in a subset \( H \) of a group \( G \), every element of \( H \) should be able to produce an inverse that is also in \( H \). For instance, consider an element \( a = 2^n 3^m \) within the subset \( H = \{ 2^n 3^m : n, m \in \mathbb{Z} \} \). The inverse would be \( a^{-1} = 2^{-n} 3^{-m} \).Since \( a^{-1} \) also follows the structure of \( 2^x 3^y \), where \( x \) and \( y \) are integers (namely \( x = -n \) and \( y = -m \)), it is evident that this inverse is within \( H \). This satisfies the inverse condition essential for \( H \) to be a subgroup of \( G \), completing the subgroup criteria.