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Is $S_4$ a group of size four and $X={1,2,3,4}$ a set of size four? Need clarification on this.

just wan to be clear on the sizes of the group and the set.

Ittay Weiss
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Lynnie
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1 Answers1

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The order of $S_4$ is $4!$ ("order" is the terminology used for "cardinality" when referring to groups). The size (or, I prefer, cardinality) of $X$ is $4$.

Does that help?

Amitesh Datta
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  • How can i found the orbit of the group on these of the question above,, i know that the orbit is one since they have the same cardinality... can you explain further? – Lynnie Apr 25 '14 at 07:07
  • Dear @Lynnie, thanks very much for your comment, although I am not sure I understand your question. Could you please clarify? (E.g., how is the action in question defined, what exactly do you want to know about the orbit etc.) If you are referring the standard action of $S_4$ on $X$, then the orbit is $X$ itself and the cardinality of the orbit is $4$. – Amitesh Datta Apr 25 '14 at 07:13
  • By the way you did answer my first question abou the size but now just want to know how to find the orbit of the group S_4 on the set x={1,2,3,4} and the stabilizer. I know that the obit will be one since they have the same cardinality but do not know how to show the stabilizer. – Lynnie Apr 25 '14 at 07:41
  • The action is not define so i'll assume its the standard action of s_4 on the set X. – Lynnie Apr 25 '14 at 07:43
  • Can i say 1={1,2,3,4}={1,2,3,4}. – Lynnie Apr 25 '14 at 07:44
  • The orbit (of any point) is $X$. The stabiliser of (say) $1$ is the subgroup of $S_4$ consisting of all permutations of ${2,3,4}$ - i.e., it is isomorphic to $S_3$. Can you see why? – Amitesh Datta Apr 25 '14 at 07:59