Roots of unity form a group. The n th root of unity is

defined by

zn=1 where n is a positive integer.

This equation has n solutions which are complex

in general.

example:

z3=1 has the solution--> 1, omega, omega^2

n roots form a group written as zn.

When n=2, the roots are real, (+1,-1)

This set forms a group under ordinary multiplication.

+1 and -1  can therefore be used to label quantum states.

You can ask how to label quantum states ?

Consider an operator (say) P, such that P2=1,

then demand that your quantum states are eigenstates

of P. The eigenvalues are either +1 or -1.

Note that this is  a multiplicative quantum number.

Roots of unity form a group. The n th root of unity is

defined by

zn=1 where n is a positive integer.

This equation has n solutions which are complex

in general.

example:

z3=1 has the solution--> 1, omega, omega^2

n roots form a group written as zn.

When n=2, the roots are real, (+1,-1)

This set forms a group under ordinary multiplication.

+1 and -1  can therefore be used to label quantum states.

You can ask how to label quantum states ?

Consider an operator (say) P, such that P2=1,

then demand that your quantum states are eigenstates

of P. The eigenvalues are either +1 or -1.

Note that this is  a multiplicative quantum number.

This PDF www.phys.nthu.edu.tw/~class/Group_theory/Chap%201.pdf does a pretty good job of explaining symmetries in general, and it does talk about the Z_2 symmetry in much the same way as Biswajoy has, above.

But these are explanations of the symmetry concept from a very formal level. In practice, in particle physics, we imagine that each particle is defined by its set of quantum numbers - weak isospin, hypercharge, colour charge, lepton/baryon number, etc. These quantum numbers represent how the object transforms under respective symmetries, and according to Noether's Theorem, for every symmetry in nature, there is a corresponding conservation law - in this case, we can think about it as each of these symmetries imply the conservation of the respective quantum numbers for every particle interaction.

In particle physics, the interactions of particles are represented as a Lagrangian. Each Lagrangian term must therefore respect ALL of the symmetries. A simple example would be electric charge - the sum of all of the electric charges of all the particles prior to a particle interaction has to equal the sum of all the electric charges of all the particles after the interaction.

The Z_2 symmetry is not an additive symmetry, but multiplicative. So, in this case, the neutral or even state is denoted as +1, and the odd state is denoted as -1. Let's say we have two particles with Z_2 states A and B, each can be either +1 or -1. A*B can only have two possibilities, too: +1 or -1! (+1*+1 = -1*-1 = +1, +1*-1 = -1*+1 = -1) So let's say that A and B interact to form two new particles, C and D. Well, we know by the conservation law that A*B = C*D. This limits what the properties of C and D are.

So this Z_2 symmetry is probably about the simplest symmetry we can define that has a corresponding conservation law. In practice, we use this Z_2 symmetry to very easily explain why something very heavy might also be stable. If we have a Z_2 symmetry, and all of the particles in the SM are in the even state (+1), and the lightest new particle that we propose in our Beyond the SM model is in the odd state (-1), then there is no kinematically accessible way for this particle to decay, because it starts off with a charge of -1, and the multiplication of the states of the particles that it decays to must also equal -1. The only way for that to happen is to have it decay to -1*+1, but there is nothing with a -1 charge lighter than this particle, so it can't happen!

So, in practice, Z_2 symmetries are a very easy way to include dark matter in a new model. That doesn't mean it is the only way, but it is a very convenient way to explain dark matter.

Z2 symmetry is a very useful to build models of dark

matter. There is one possible problem in this case

which one should bear in mind. If a Z2 symmetry is

spontaneously broken then the vacuum is  two fold

degenerate. At the bottom of the Mexican hat potential

(for a scalar) which breaks Z2 symmetry there are

two points giving two different vacuum. Then there

is a minimum energy field configuration which interpolates

between two different vacua. This is a domain wall.

Non observation of domain walls put constraints

on the model building process. One way to evade

this problem is to break the discrete symmetry at

a high scale more than 1012 GeV or so. Then

domain wall problem will be diluted by infationary

expansion of the Universe.

Similar domain walls are studied in laboratory

for microscopic systems also. Here is a picture.