I'll explain in terms of the field of mathematics that dominated until modern times, EUCLIDEAN GEOMETRY (thought to be "the only geometry" until creation of The Non-Euclidean Geometries in the 19th century). Given the GROUP concept, you can encapsulate EG in a single sentence: "EUCLIDEAN GEOMETRY is the set of all properties CONSERVED BY THE EUCLIDEAN GROUP."
What's the EUCLIDEAN GROUP?
- Start with the PRIMARY PROPERTY of EUCLIDEAN GEOMETRY, namely, CONGRUENCE: EQUIVALENCE OF SHAPE AND SIZE (LENGTH AND ANGLE).
- WHAT TRANSFORMATIONS CONSERVED CONGRUENCE? TRANSLATION, ROTATION, REFLECTION (Shove, Turn, Flip). Given a structure, translate it in space, resulting in a structure that EXACTLY MATCHES the original. Same for ROTATION (shove) and REFLECTION (flip). And we can "combine" these TRANSFORMATIONS, say, TRANSLATION FOLLOWED BY ROTATION, or vice versa. (In mathematics, "followed by" is CONCATENATION.)
- CONCATENATION OF ANY NUMBER OF SUCH TRANSFORMATIONS IS EQUIVALENT TO JUST ONE OF THEM!
- THEN THE EUCLIDEAN GROUP IS A SET CLOSED UNDER TRANSLATION, ROTATION, REFLECTION, AND THEIR CONCATENATIONS. CONGRUENCE IS CONSERVED UNDER THESE, so we make this THE CRITERION for all the "desirable" properties. For example, the geometric property of AREA is CONSERVED UNDER THE EUCLIDEAN GROUP, hence, it is part of the PRIMARY STUDY of EUCLIDEAN GEOMETRY. Repeating: EUCLIDEAN GEOMETRY IS THE STUDY OF ALL PROPERTIES CONSERVED UNDER THE EUCLIDEAN GROUP.
Similarly, the mathematical physicist can encapsulate Einstein's Special Theory of Relativity via "group": SPECIAL RELATIVITY IS THE STUDY OF ALL PROPERTIES CONSERVED UNDER THE LORENTZ GROUP OF TRANSFORMATIONS.
In ARITHMETIC, THE RATIONAL NUMBER SYSTEM FORMS AN ADDITIVE GROUP (CLOSED UNDER ADDITION AND SUBTRACTION), AND RATIONALS WITH ZERO REMOVED FORM A MULTIPLICATIVE GROUP (CLOSED UNDER MULTIPLICATION AND DIVISION). The previous statement is ENCAPSULATED by the sentence: "THE RATIONAL NUMBERS FORM A FIELD."
The American mathematical physicist, Murray Gell-Mann, noticed that several short-living PARTICLES had similar PROPERTIES so that one might think of them as being TRANSFORMED INTO ONE ANOTHER by a PHYSICAL GROUP. He organized them into a structure he amusingly called "The Eight-Fold Way" (for the injunctions of Buddha). But the structure was INCOMPLETE. Using the GROUP idea, Gell-Mann predicted what the properties of this particle would be. Later on, this particle was discovered, with the properties Gell-Mann predicted. So, in this, and in other cases, the GROUP-concept has PREDICTIVE POWER.
The basic concept behind GROUP is SYMMETRY, the GREAT CONCEPT OF ART.
The German-American mathematician, Emmy Noether (1882-1935), used the GROUP idea to show that GROUP IS BEHIND EVERY LAW OF NATURE. (Great enough for you?)
As some of you may know, our word "group" derives from the French word "groupe" which was brought into Mathematics by the amazing boy, Évariste Galois (1811-1832), who made GROUP THEORY popular by using it to PROVE THAT 5TH DEGREE ALGEBRAIC EQUATIONS EXISTS WHICH CANNOT BE SOLVED BY RADICALS ("root extraction"). Galois did this work at the age of 16! (No genius in any field -- not even Mozart -- accomplished so much at such a young age!) Only to be killed in a duel at age 20. The MODULAR GROUPS he developed at this time provides the ECONOMICAL CODING USED FOR SATELLITE SIGNALS FOR OUR COMMUNICATION AND ENTERTAINMENT.