Checklist

For all authors... (a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? If you used crowdsourcing or conducted research with human subjects... (a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] (b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] (c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? A.1 Group, Coset and Quotient Space A group G is a set of elements equipped with a binary operation (denoted as) that satisfies the following group axioms: 1. (Closure) For all a, b G, a b G. 2. (Associative) For all a, b, c G, (a b) c = a (b c). 3. (Identity element) There exists an identity element e in G such that, for any a G we have e a = a e = a. 4. (Inverse element) For each a G, there exists an element b G such that a b = b a = e where e is the identity element. The centered dot can sometimes be omitted if there is no ambiguity. In this work, we are mainly interested in symmetry groups where each group element is associated with a symmetry of a pattern, which is a transformation that leaves the pattern invariant.