Endliche Gruppen I
01 Jan 1967-
About: The article was published on 1967-01-01. It has received 5518 citations till now.
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TL;DR: Wiles as discussed by the authors proved that all semistable elliptic curves over the set of rational numbers are modular and showed that Fermat's Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.
Abstract: When Andrew John Wiles was 10 years old, he read Eric Temple Bell’s The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat’s Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that an + bn = cn. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.
1,822 citations
29 Jun 1997
TL;DR: In this article, the problem of finding quantum error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a trace inner product.
Abstract: The unreasonable effectiveness of quantum computing is founded on coherent quantum superposition or entanglement which allows a large number of calculations to be performed simultaneously. This coherence is lost as a quantum system interacts with its environment. In the present paper the problem of finding quantum-error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.
1,525 citations
01 Jan 1995
1,303 citations
Book•
01 Jan 1994TL;DR: In this article, the authors introduce the special odd case General lemmas Theorem (C^*_2$): Stage 1 Theorem $C^*) 2$: Stages 2 and 3 IV$_K$: Preliminary properties of $K$-groups Background references Expository references Glossary Index
Abstract: General introduction to the special odd case General lemmas Theorem $C^*_2$: Stage 1 Theorem $C^*_2$: Stage 2 Theorem $C_2$: Stage 3 Theorem $C_2$: Stage 4 Theorem $C_2$: Stage 5 Theorem $C_3$: Stage 1 Theorem $C_3$: Stages 2 and 3 IV$_K$: Preliminary properties of $K$-groups Background references Expository references Glossary Index.
893 citations
Cites methods from "Endliche Gruppen I"
...For example, the subgroups of PSL2(q) are determined in [Hu1], the Bender-Thompson odd prime signalizer lemma is proved in [HuB1], and Glauberman’s Z∗-theorem in [F1]....
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TL;DR: In this paper, the maximal subgroups are studied up to conjugation, up to the case of finite subgroups, where the subgroup M of G is a maximal subgroup of G and there exists no subgroup H such that H < G.
Abstract: (1.1) Definition Let 1 6= G be a group. A subgroup M of G is said to be maximal if M 6= G and there exists no subgroup H such that M < H < G. IfG is finite, by order reasons every subgroupH 6= G is contained in a maximal subgroup. If M is maximal in G, then also every conjugate gMg−1 of M in G is maximal. Indeed gMg−1 < K < G =⇒ M < g−1Kg < G. For this reason the maximal subgroups are studied up to conjugation. (1.2) Lemma Let G = G′ and let M be a maximal subgroup of G. Then:
654 citations