Let H be a subgroup of a group G. Prove that the number of conjugates of H in G is |G: N(H)|. (T

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Let H be a subgroup of a group G. Prove that the number of...

Key Concepts

Index of a Subgroup

The index of a subgroup H in a group G, denoted |G : H|, is defined as the number of distinct left (or right) cosets of H in G. It provides a measure of how H fits into the larger structure of G. In the context of counting conjugates, the index |G : N(H)| indicates the number of distinct conjugates since each distinct coset corresponds to a unique conjugate subgroup under the conjugation action.

Conjugation Action

The conjugation action of a group on itself or on its subgroups is defined by taking each element g in G and mapping a subgroup H to gHg?¹. This action partitions the set of subgroups into orbits, where each orbit consists of all conjugate subgroups. It is the fundamental mechanism by which group symmetries are explored, particularly in understanding how subgroups relate under the structure of the whole group.

Normalizer of a Subgroup

The normalizer of a subgroup H in G, denoted N(H), is the set of elements in G that stabilize H under conjugation; that is, g in G such that gHg?¹ equals H. The normalizer is itself a subgroup of G and measures the 'degree of symmetry' of H within G. In the context of conjugation, it provides a natural way to classify the distinct conjugate subgroups since elements outside the normalizer yield genuinely different conjugates.

Orbit-Stabilizer Theorem

The Orbit-Stabilizer Theorem is a result in group theory that relates the size of a group acting on a set to the sizes of the orbits and stabilizers of elements in that set. When G acts on its subgroups by conjugation, the orbit of H consists of its conjugates, and the stabilizer of H under this action is precisely the normalizer N(H). The theorem then implies that the number of conjugates of H, which is the size of its orbit, equals the index of its normalizer, |G : N(H)|.

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