MIDTERM 1 FOR ALGEBRA
Date: 1999, October 25, 10:10–11:00AM Each Problem is worth 15 points.
1. LetF be the set of all real-valued functions having the setRof all real numbers as domain. Define the operation “◦” onF by (f◦g)(x) =f(g(x)) forf, g∈F. Show that the operation “◦” is associative.
2. Let hS,∗i, hS0,∗0i hS00,∗00i be three sets with binary operations. Suppose that φ: S → S0 and ψ:S0→S00are both isomorphisms. Show thatψ◦φis an isomorphism ofhS,∗i andhS00,∗00i.
3. LetGbe a group and letabe a fixed element ofG. Show that the subset Ha ={x∈G|xa=ax}
is a subgroup ofG.
4. LetH be the subgroup ofZ30 generated by 25. Find|H|.
5. Letσ= (1245)(36) inS6. Find the index ofhσiin S6.
6. LetGbe a group of orderpq, wherepandqare prime number. Show that every proper subgroup of Gis cyclic.
7. LethG,· ibe a group. Consider the new binary operation∗ onGdefined by a∗b=b·a
fora, b∈G. Then hG,∗i is a group (you don’t need to check this). Show that hG,∗i is isomorphic to hG,· i.
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