Download e-book for kindle: A first course in abstract algebra: rings, groups, and by M. Anderson

By M. Anderson

ISBN-10: 1584885157

ISBN-13: 9781584885153

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6. ,*fc. 3 tell us that EH is an injective homomorphism of the algebra F(X) into the product algebra C(X)J that establishes an algebraic and order-theoretic isomorphism of F(X) with its image I(X). For all r € J, the projection of F(X) into the r t h coordinate space of C(X)J is a surjection. In fact, Vr € J, Dl(fi)(r) maps the subset C(X) of F(X) bijectively onto the r t h coordinate space C(X) : Let ji e C(X). , M(/i)(r) = ft. 34 Recall that an algebra A is said to be a subdirect product of a family of algebras {Ab \ b E B}, where B is an arbitrary index set, if A is isomorphic to a subalgebra of the product algebra U{Ab \ b e B} with the property that its projection into each coordinate space Ab is a surjection.

4) Since 1 W / ^ (W( x( ^W x )_"11)) = M /xW(x-V) MC*" M(x) -= M MW( (*M) *W) M (* _ V) -= M ( * _ ) = MOO V x G G and xx(*)((x/x)c 3 (w)) > = /x<*)(xy/x) /x(xy) /x(x) A /x(y) /i(y) //x(*)(x,x)A/x(*)(y/x) x^(x/x)A/xW « V x, y G G, it follows that //*) G L(G//x). Moreover, since /x(*)((xxx)c //*) ((x/x) o5 (y/x)) {vv)) = = = = = /x(*)(xy/x) /x<*>(xy/x) /x(xy) /^y) /x(yx) /*(y*) fi^(yxfi) n{*\yxfi) /x(*)((^)o (x/x)) /x(*)((y/x)o(x/x)) V x, y G G. We conclude that /x<*) G A/X(G//x). 11 is called the quotient group (or factor group) of G relative to the normal L-subgroup /x.

F) If G is as in (d) and is Abelian, then \x is a direct product of fuzzy subgroups. (g) If G is as in (d) and is cyclic, then \i is a direct product of fuzzy subgroups of Sylow subgroups of G. (h) If G is the symmetric of degree 3, determine the cardinality of L{G)/ ~ . 3. (19) Suppose that G is a finite group. Prove the following statements: (a) G is Dedekind if and only if every fuzzy subgroup of G is normal. (b) [35] Suppose that G is of order p\p2 • • -Pn? the p^s being primes, but not necessarily distinct.

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A first course in abstract algebra: rings, groups, and fields by M. Anderson

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