Amalgamated products

An amalgamated product is a mathematical construction where two groups are combined by identifying a common subgroup. Think of it as merging two structures along shared parts, creating a larger group that contains both original groups while respecting their shared features. This process can be visualized as gluing two puzzle pieces together along matching edges, ensuring the connection is consistent. Amalgamated products are useful in abstract algebra for building complex groups from simpler ones, and they help in understanding how different algebraic structures relate and combine within the broader mathematical framework.