2018年3月31日 · Quoting from Wolfram definition of group action -. In general, a group action is when a group acts on a set, permuting its elements, so that the map from the group to the permutation group of the set is a homomorphism.

2021年2月6日 · Group Action from to Subgroup to Group: Care in Differentiating Between Binary Op., Group Action Op. 4. ...

2019年10月22日 · The notion of (abstract) group action arises as soon as you move from the original notion of "group" as group of permutations (which natively fulfills the conditions $\iota(s)=s, \forall s\in S$, and $\sigma(\tau(s))=(\sigma\tau)(s),\forall s\in S, \forall \sigma,\tau\in \operatorname{Sym}(S)$) to the one of abstract group.

2015年9月6日 · Orbits of the group are precisely sets of points that you can reach with your group action from some starting point. Hence transitivity means that there is only one orbit. For example if you consider group of rotations with respect to fixed point acting on Euclidean space, orbits are just spheres of constant radius with respect to that point.

2017年1月26日 · $\begingroup$ by proving $\lambda$ to be bijective and group preserving operation implying homomorphism .and conversely if $\lambda$ is a group homorphism then by defining a group action and proving that its well defined . $\endgroup$

2017年11月9日 · Group Action from to Subgroup to Group: Care in Differentiating Between Binary Op., Group Action Op. 2 What is a "field action", and how does it relate to group actions?

2016年10月17日 · For an abelian group we just get a string of ones. Functions that are constant on conjugacy classes are called class functions. As an example there's characters. They were important in Feit-Thompson's theorem, leading to the classification of finite simple groups. One of the most basic types of group action is conjugation by an element of the ...

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2015年1月14日 · On the other hand, I am a big fan of Chapter 1 of the book "Combinatorial group theory" by Magnus, Karrass and Solitar (£9.99, second hand, amazon.co.uk). It is, perhaps, the ultimate in introductions to presentations. It is also one of the classic texts in geometric and combinatorial group theory. $\endgroup$ –

2020年9月26日 · The notion of abstract group action is patterned upon the natural action of the set ...