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Thus t2 . . tn sn−1 . . s2−1 ∈ H2 (e)H3 (e) . . Hn (e). By (B3) we have t1−1 s1 = t2 . . tn sn−1 . . s2−1 = e. Therefore, s1 = t1 . Similarly we can show si = ti for every i = 2, . . , n. Consequently we obtained (A2). In group theory, the external direct product G = G 1 × G 2 always admits an internal direct decomposition of its subgroups isomorphic to G 1 and G 2 . Let H1 be {(g1 , 1) | g1 ∈ G 1 } and H2 be {(1, g2 ) | g2 ∈ G 2 }, respectively. Then G is the internal direct product of H1 and H2 .

Ii) Let R be a commutative PID. Assume that M is a nonzero semisimple Rmodule with nonzero annihilator in R. 3, M has only a finite number of homogeneous components. Let {Hk | 1 ≤ k ≤ s} be the set of all homogeneous components of M. For k, 1 ≤ k ≤ s, we put Hk = ⊕α M(k,α) with each M(k,α) simple. So M(k,α) ∼ = R/pk R for k, 1 ≤ k ≤ s, with pk a nonzero prime. We put Pk = AnnR (Hk ) for k, 1 ≤ k ≤ s. Then Pk = pk R. For a nonnegative integer , we can routinely verify that Pk− = (1/pk )R for k, 1 ≤ k ≤ s.

Note that E(S) is isomorphic to the largest band image of S and E(S) ∼ = S/H. A nonempty subset of an orthocryptogroup S is called a sub-orthocryptogroup if it forms an orthocryptogroup under the multiplication of S, that is, a nonempty subset is a sub-orthocryptogroup if and only if it is closed under taking an inverse and multiplication. Suppose S is an orthocryptogroup and φ is the natural homomorphism of S onto the largest band image B, that is, B ∼ = S/H. A sub-orthocryptogroup H of S is called full if E(H ) = E(S).

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An Introduction to Wavelets Through Linear Algebra Solutions Manual by Frazier M.

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