Algebraic Structures [Lecture notes] by Thomas Markwig Keilen

By Thomas Markwig Keilen

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Extra resources for Algebraic Structures [Lecture notes]

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7. Since disjoint cycles commute with each other we may assume without loss of generality 15 that k1 ≥ k2 ≥ . . ≥ kt. 15 “We may assume without loss of generality” in principle means that we only consider a special case and prove this one case. However, the other cases will follow in the same way. Doing only the 38 Moreover, since σ is not the neutral element id we know that k1 ≥ 2 and that the number s = max{r | 1 ≤ r ≤ t, kr ≥ 2} is defined. Thus σi = id for i = s + 1, . . , t and σ = σ1 ◦ .

N}) to be the identity. 6 The permutations σ= 1 2 3 4 4 1 3 2 ∈ S4 and π= 1 2 3 4 5 4 1 3 2 5 ∈ S5 34 are all 3-cycles which permute the numbers 1, 4, 2 cyclically. In the above notation they are thus described as and σ = (1 4 2) π = (1 4 2). This shows the disadvantage of the new notation. It does not give any information on the set of definition of the permutation. Different maps are represented by the same scheme. However, we are willing to pay this price for a representation which is both shorter and better organised.

We say that C detects errors of Type I if and only if (g1, . . , gn)t ∈ C and gi′ ∈ G with gi′ = gi implies that (g1, . . , gi−1, gi′ , gi+1, . . , gn)t ∈ C. c. We say that C detects errors of Type II if and only if (g1, . . , gn)t ∈ C with gi = gi+1 implies that (g1, . . , gi−1, gi+1, gi, gi+2, . . , gn)t ∈ C.

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