Learning groups can be daunting. One guide is study each mentioned group. Study how symmetry is established in each group.

This article may help it covers the Poincare group with key details into the Lorentz group (SO)1.3 Hint one is time dimension 3 is spatial dimensions...

O means orthogonal

S is special.

This is included in the article. The lorentz group is needed to understand the Poincare group. So study both...

Put simply a rotation is the spatial components but doesn't involve time. A boost involves the spacetime translations. The Lorentz group has three rotations and 3 boosts. Each is listed in the article.

This article also shows spin statistics aspects of the Poincare group.

Representations of the Symmetry Group of Spacetime

http://www.google.ca...4mg7ARF7VO4hXKw
Unfortunately it doesn't cover parity well.

PS the wiki Lorentz group page is fairly decent.

oh almost forgot you will need to familiarize yourself with Bra-ket notation. The article uses it. group theory often use this notation and others.

https://en.m.wikiped...–ket_notation
YdoaPs, Migl, Studiot and wtf have all provided several key details. So study carefully what each is saying. Keep at it.

You will be amazed how interconnected particle physics, quantum physics, thermodynamics and GR truly is. All described by geometry under the

SO(3)×SO(2)×U(1). standard model particles with no supersymmetric particles. (I don't believe Pati-Salam SO (6) which deals with left and right handedness falls under this but some aspects do.) Particularly in U(1) and SO (2) however Im not positive on Pati-Salam I studied it under the SO (10) MSM and MSSM.

lol I'm fairly confident I lost the audience on that last comment... Just thinking to myself as to group representation under SO (1.3) for left/right hand chirality ( Grr thanks Studiot)

Mordred mentioned rotations.

There are also what are known as 'forbidden' rotations' we are important for non symmetrical (handed or chiral) objects.

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if I recall correct Pati-Salam SO(4) included SO (3) left and SO (3) right. but not positive lol. Thankfully "Pati-Salam A la SO(10) on arxiv will tell me.

For others reading an allowable rotation or boost must fall within a symmetry group. As Studiots comment indicates some rotations or boosts may fall into different groups or sub groups. Adding a rotation, boost or translation not already described under a group, changes to a new group...

** key note, study symmetry relations in just the spatial components, before tackling the boosts due to time components ** this is key to understanding the Minkowskii metric, or any other GR metric.

aka line element to ds^2 in regards to Pythagoras theory

**Edited by Mordred, 1 September 2016 - 07:28 AM.**