By John B. Fraleigh

A well-known publication in introductory summary algebra at undergraduate point.

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**Extra resources for A First Course in Abstract Algebra, 7th Edition**

**Example text**

1. Endomorphism Algebras. 3), the description of simple objects of FD,R can be made much more precise. 2. Proposition : Let R be a commutative ring with identity, and let D be an admissible subcategory of the biset category C. If G is an object of D, denote by IG the R-submodule of EndRD (G) generated by all endomorphisms of G which can be factored through some object H of D with |H| < |G|. Then IG is a two sided ideal of EndRD (G), and there is a decomposition EndRD (G) = AG ⊕ IG where AG is an R-subalgebra, isomorphic to the group algebra ROut(G) of the group of outer automorphisms of G.

10, by exchanging the positions of G and H, and taking for U the (G, H)-biset IndG H . Recall that this is the set G itself, with biset structure given by left multiplication in G and right multiplication by elements of H. So U is both left-free and left-transitive, so the maps αU,X and βU,Y are isomorphisms. 14, by considering the opposite bisets, and observing that the (G, G)-bisets X op and X are isomorphic. Similarly, Assertion 4 and the ﬁrst isomorphism of Assertion 3 follow from the case H = G/N , and U = Def G G/N .

In particular, the ghost map Qφ : QB(G) → H∈[sG ] Q is an algebra isomorphism, where QB(G) = Q ⊗Z B(G). This shows that QB(G) is a split semi-simple commutative Q-algebra, whose primitive idempotents are indexed by [sG ]. 2. Theorem : Let G be a ﬁnite group. If H is a subgroup of G, denote by eG H the element of QB(G) deﬁned by eG H = 1 |NG (H)| |K|μ(K, H) [G/K] , K≤H where μ is the M¨ obius function of the poset of subgroups of G. G Then eG = e H K if the subgroups H and K are conjugate in G, and the G elements eH , for H ∈ [sG ], are the primitive idempotents of the Q-algebra QB(G).