What is the meaning of symmetry of equalities?

Question

For the equation:

$$a+\frac1b=b+\frac1c=c+\frac1a=t$$

According to @FundThmCalculus's Answer We have symmetry of equalities. I wonder what does that mean. in other word when equations are symmetric in mathematics?

Answer 1

This is known to some of us as a cyclic symmetry, with the group of symmetries being $A_3$ (also the cyclic group $C_3$, but here named as a normal subgroup of $S_3$, the full group of symmetries on three objects).

The rest of this answer probably takes you well away from the context of your original problem, but it does attempt to illustrate something of mathematical symmetry - which is expressed very often in group theory.

Full symmetries and cyclic symmetries remain interesting as the number of variables increases, but the algebraic study of symmetry (study of algebraic symmetries) is founded more naturally around Galois theory. Sometimes the full group of symmetries will be involved, and the study of symmetric functions of $n$ variables invariant under the full group is a place to begin. Every (suitably defined) symmetric function can be expressed in terms of certain elementary symmetric functions - and that is often useful to know when embarking on practical calculations, though cases with three variables are soon learnt. It is as the number of variables becomes larger that things start to get interesting. Here is a rough outline of the kind of thing which can happen.

Many systems $S$ are symmetric under subgroups of the symmetric group (rather than fully symmetric under the whole symmetric group - cyclic symmetry is an example of this), and we sometimes want to analyse subsystems $T$ of $S$. Galois theory exploits the fact that, if $T$ sits nicely in $S$, very often normal subgroups will be involved when we examine the relevant groups of symmetries. In fact for each normal subgroup of an appropriate $S$ we hope to find a nice sub-system $T$, and each nice subsystem will relate to a normal subgroup.

You might want to look at the process of solving cubic and quartic equations in a single variable, which is where this starts to take root. Note, for example, that the three expressions $ad+bc, ac+bd, ab+cd$ are permuted amongst themselves under any permutation of the four letters $a, b, c, d$. It is this symmetry which is essentially exploited in a key step of solving the quartic as it takes us from the original four letters (the roots of the equation) down to three expressions. And this in turn relates to the fact that the symmetric group on four letters $S_4$ has a normal subgroup $V$ of order $4$.

The fact that the quintic equation relates to the symmetric group $S_5$, together with the fact that the alternating group $A_5$ is simple, means that $S_5$ has no small normal subgroups, and that means there are no sub-systems of roots small enough to make the same kind of reduction as in the case of the quartic. Thus we have the famous Abel-Ruffini Theorem that the quintic equation cannot be solved "by radicals". Obviously there is a fair amount of detail missing, but it is an illustration of how exploiting our knowledge of symmetry can show us what is, and what is not possible.