# Why $O_i$'s are blocks in this group action?

I was thinking about the problem from Dummit-Foote (Art. 4.1 Question 9)

N.B. The definition of block as given in Ex. 7: Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ such that for all $\sigma\in G$ either $\sigma(B) = B$ or $\sigma(B)\cap B = \emptyset$.

I'm not looking for a solution. I did it for the first part. What I want to know is that why $O_i$'s are blocks? $G$ is not necessarily a permutation group. Do I need to consider the effect of the group $\{\sigma_g:A\to A:a\mapsto ga\}$ instead of $G?$

Please clarify and help me. By the way I would like to have an elaborate answer as I'm self-studying Group action for the first time.

Given an action of a group $G$ on a set $A$, we can define the permutation representation of $G$. More specifically, define $\rho:G\to S_A$ by $\rho(g)(a)=g\cdot a$. This map might not be injective, but if the action is transitive, then $\operatorname{im}G$ is now a transitive permutation group on the finite set $A$.
Now, define the blocks of $G$ acting on $A$ to be those of the transitive permutation group $\operatorname{im}G$ acting on $A$ (using the definition you've listed). With this definition, it follows that the $\mathcal{O}_i$ are blocks under the action of $G$, because they are blocks under the action of $\operatorname{im}G$ with your original definition.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9858627319335938, "perplexity": 40.327628235218086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496668772.53/warc/CC-MAIN-20191116231644-20191117015644-00021.warc.gz"}