By A.I. Kostrikin, I.R. Shafarevich, E. Behr, Yu.A. Bakhturin, L.A. Bokhut, V.K. Kharchenko, I.V. L'vov, A.Yu. Ol'shanskij

The algebra of sq. matrices of dimension n ~ 2 over the sphere of complicated numbers is, obviously, the best-known instance of a non-commutative alge 1 bra • Subalgebras and subrings of this algebra (for instance, the hoop of n x n matrices with critical entries) come up certainly in lots of parts of mathemat ics. traditionally even though, the research of matrix algebras used to be preceded by way of the invention of quatemions which, brought in 1843 through Hamilton, came across ap plications within the classical mechanics of the previous century. Later it became out that quaternion research had vital purposes in box idea. The al gebra of quaternions has develop into one of many classical mathematical gadgets; it truly is used, for example, in algebra, geometry and topology. we are going to in short concentrate on different examples of non-commutative earrings and algebras which come up obviously in arithmetic and in mathematical physics. the outside algebra (or Grassmann algebra) is wide-spread in differential geometry - for instance, in geometric conception of integration. Clifford algebras, which come with external algebras as a distinct case, have functions in rep resentation thought and in algebraic topology. The Weyl algebra (Le. algebra of differential operators with· polynomial coefficients) frequently seems to be within the illustration concept of Lie algebras. in recent times modules over the Weyl algebra and sheaves of such modules grew to become the root of the so-called microlocal research. the speculation of operator algebras (Le.

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Resp. H(i, n) = {f ∈ Sn ; f (i) = n}, where 1 ≤ i ≤ n. 2. If G = Z and H = Zn, n ≥ 1, there are again n cosets, namely the sets r + Zn, 0 ≤ r < n. So a coset consists of all integers which give after division with n the same remainder r, 0 ≤ r < n. 3. A geometric interpretation: Let G = R2 and H ≤ R2 a line through the origin. The cosets mod H are then nothing but the lines parallel to H. 4. The cosets of S1 ≤ C∗ are the circles centered at the origin. 28. Let H ≤ G be a subgroup of the group G.

From that and A2 = −E = B 2 we easily derive that {±E, ±A, ±B, ±C} with C := AB is a group isomorphic to the quaternion group Q. First of all AB, BA ∈ Im(H) = E ⊥ , since for example σ(AB, E) = σ(A, B ∗ ) = −σ(A, B) = 0. Hence AB = −(AB)∗ = −B ∗ A∗ = −(−B)(−A) = −BA. Furthermore, AB is orthogonal to both A and B: σ(AB, B) = σ(A, BB ∗ ) = σ(A, E) = 0, while σ(AB, A) = σ(A, AB ∗ ) = σ(AB ∗ , A) = σ(−AB, A) = −σ(AB, A), whence σ(AB, A) = 0 as well. As a consequence of this discussion we see that the product of quaternions is closely related to vector geometry in three space: The map 1 Im(H) × Im(H) −→ Im(H), (A, B) −→ Im(AB) = (AB − BA) 2 is nothing but the vector product of the “vectors” A, B ∈ Im(H) ∼ = R3 : It is bilinear, alternating and associates to two orthogonal unit vectors a unit vector orthogonal to both factors.

We forget about the condition that a composition series be strictly increasing and prove the statement for normal series with simple factors, but of given length - we may extend the shorter series by adding terms {e} or G. This gives our theorem, since the trivial group then has the same multiplicity in both sequences. Assume first r = s = 2. Then our composition series are of the form {e} ⊂ E ⊂ G and {e} ⊂ F ⊂ G with simple normal subgroups E, F ⊂ G and simple G/E, G/F . If G itself is simple or F = E, nothing remains to be shown.