Janko group
Encyclopedia
In mathematics, a Janko group is one of the four sporadic simple
group
s named for Zvonimir Janko
. Janko constructed the first Janko group J1 in 1965. At the same time, Janko also predicted the existence of J2 and J3. In 1976, he suggested the existence of J4. J2, J3, and J4 were all later shown to exist.
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
s named for Zvonimir Janko
Zvonimir Janko
Zvonimir Janko is a Croatian mathematician who is the eponym of the Janko groups, sporadic simple groups in group theory.Janko was born in Bjelovar, Croatia. He studied at the University of Zagreb where he received Ph.D. in 1960. He then taught physics at a high school in Široki Brijeg in Bosnia...
. Janko constructed the first Janko group J1 in 1965. At the same time, Janko also predicted the existence of J2 and J3. In 1976, he suggested the existence of J4. J2, J3, and J4 were all later shown to exist.
Janko groups
- The Janko group J1Janko group J1In mathematics, the smallest Janko group, J1, is a simple sporadic group of order 175560. It was originally described by Zvonimir Janko and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century...
has order 175 560 = 23 · 3 · 5 · 7 · 11 · 19. It is the only Janko group whose existence was proved by Janko himself. - The Hall–Janko groupHall–Janko groupIn mathematics, the Hall-Janko group HJ, is a finite simple sporadic group of order 604800. It is also called the second Janko group J2, or the Hall-Janko-Wales group, since it was predicted by Janko and constructed by Hall and Wales...
has order 604 800 = 27 · 33 · 52 · 7. It is also known as J2, HJ, or the Hall–Janko–Wales group. It was constructed by Marshall Hall, Jr.Marshall Hall (mathematician)Marshall Hall, Jr. was an American mathematician who made contributions to group theory and combinatorics.- Career :...
and David Wales. - The Janko group J3Janko group J3In mathematics, the third Janko group J3, also known as the Higman-Janko-McKay group, is a finite simple sporadic group of order 50232960. Evidence for its existence was uncovered by , and it was shown to exist by...
has order 50 232 960 = 27 · 35 · 5 · 17 · 19. It is also known as the Higman–Janko–McKay group. It was constructed by Graham HigmanGraham HigmanGraham Higman FRS was a leading British mathematician. He is known for his contributions to group theory....
and John McKay. - The Janko group J4Janko group J4In mathematics, the fourth Janko group J4 is the sporadic finite simple group of order 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 = 86775571046077562880 whose existence was suggested by Zvonimir Janko . Its existence and uniqueness was shown by Simon P. Norton and others in 1980...
has order 86 775 571 046 077 562 880 = 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43. It was constructed by Simon P. NortonSimon P. NortonSimon Phillips Norton is a mathematician in Cambridge, England, who works on finite simple groups. He constructed the Harada–Norton group, and in 1979 together with John Conway proved there is a connection between the Monster group and the j-function in number theory. They dubbed this monstrous...
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