1. The problem statement, all variables and given/known data
Let S be a set and let a be a fixed element of S. Show that s is an element of Sym(S) such that s(a)=a is a subgroup of Sym(S).
2. Relevant equations
3. The attempt at a solution
Related Calculus and Beyond Homework News on Phys.org
The Symmetric group is already known to be a group so you do not need to show associativity. In general to show a subset H is a subgroup you need:
1) The inverse is in H
2) H is closed under the group operation (a,b in h implies ab in H).
3) If a is in H then a^{-1} is in H
however 2 and 3 imply 1 so you only really need to show the last two. The one step solution is to show that a in H, b in H implies ab^{-1} is in H because that implies 2 and 3. In your case let f,g be bijections from S to S. Then show:
1) f(a) = a implies f^{-1}(a) =a
2) f(a) = a, g(a) =a implies f(g(a)) = a
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling We Value Civility
• Positive and compassionate attitudes
• Patience while debating We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving