Question

This is the preceding exercise below:

A subgroup H is conjugate to a subgroup K of a group G if there exists an inner automorphism ig of G such that ig[H] = K. Show that conjugacy is an equivalence relation on the collection of subgroups of G.

Solve the the below:

Characterize the normal subgroups of a group G in terms of the cells where they appear in the partition given by the conjugacy relation in the preceding exercise.

Answer 1

A homologous series is a sequence of organic compounds where each differs by a specific factor such as a CH2 unit, with predictable variations in properties. The Ka of an acid and the Kb of its conjugate base have a significant inverse relationship in acid-base chemistry.

Step-by-step explanation:

The concept you are referencing is from organic chemistry, specifically relating to homologous series. A homologous series is a family of organic compounds each differing by a constant unit, typically a CH2 group. Starting with propane (C3H8), each subsequent member of the series adds one methylene unit (CH2).

This series includes compounds like butane, pentane, hexane, etc., each with one more carbon atom than the previous compound. Homologs within the series display similar chemical properties that change progressively and predictably, allowing chemists to deduce characteristics of other members of the series based on known properties, such as boiling points, solubility, and reactivity.

When studying acid-base chemistry, the relationship between the acid dissociation constant (Ka) and the base dissociation constant (Kb) of conjugate acid-base pairs is significant.

The ionization of hydrocyanic acid (HCN) in water forming an acidic solution and the reaction of its conjugate base (CN−) with water to form a basic solution demonstrate this concept. The magnitudes of Ka and Kb are related in such a way that strong acids will have weak conjugate bases and vice versa.