Adelic groups are mathematical structures that arise in the field of number theory, particularly in the study of algebraic groups. They combine local and global properties of numbers by considering the behavior of a group over various completions of a number field, such as the p-adic numbers and the real numbers. This allows mathematicians to analyze properties that may not be visible when looking at the numbers in isolation. In essence, an adelic group is formed by taking the product of a group over all local fields, including both finite and infinite places. This construction helps in understanding the symmetries and structures of algebraic objects, making it a valuable tool in areas like automorphic forms and arithmetic geometry.