3. Groups

If g and h are two rotations that preserve a symmetry, then the composition of first h and then g will also be a rotation (written g * h) that preserves the symmetry. (For equations, the symmetries are not actually rotations, but for simplicity's sake I use this geometric picture.) This composition satisfies some rules:

• (g * h) * f = g * (h * f). The left hand side here is the rotation f, followed by the composition of the rotations h and g. The right hand side is first the rotation that is the composition of f and then h, followed by the rotation g. But, the result of these two methods are the same.
• We have the identity rotation 1, which leaves everything in peace. It satisfies 1 * g = g * 1 = g.
• Every rotation has an inverse rotation g^-1 that takes us back to where we were. It satisfies g * g^-1 = g^-1 * g = 1.

Such groups of symmetries can be finite, such as the symmetries of a football or Rubik's cube, or infinite, such as the symmetries of a sphere. But now, one can abstract the situation and ask oneself "which (finite) groups of symmetries are there in all?" And so opens up a whole new world for one to study. The classification of the fundamental building blocks of finite groups, which are called simple groups, was completed in the 1970s. One of those who were responsible for the biggest breakthrough in regards to this was J. G. Thompson, who was awarded the Abel prize in 2008.

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