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- In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group is said to be free-by-cyclic if it has a free normal subgroup such that the quotient group is cyclic. In other words, is free-by-cyclic if it can be expressed as a group extension of a free group by a cyclic group (NB there are two conventions for 'by'). Usually, we assume is finitely generated and the quotient is an infinite cyclic group. Equivalently, we can define a free-by-cyclic group constructively: if is an automorphism of , the semidirect product is a free-by-cyclic group. An isomorphism class of a free-by-cyclic group is determined by an outer automorphism. If two automorphisms represent the same outer automorphism, that is, for some inner automorphism , the free-by-cyclic groups and are isomorphic. (en)
- 在群論中，如果有一個自由正規子群F，使得商群G/F是循環群，则群G稱為free-by-cyclic群， 換言之，如果G是一個循環群對一個自由群的群擴張，则G是一个free-by-cyclic群。 若F是有限生成群，則稱G是(finitely generated free)-by-cyclic群。 (zh)
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- 在群論中，如果有一個自由正規子群F，使得商群G/F是循環群，则群G稱為free-by-cyclic群， 換言之，如果G是一個循環群對一個自由群的群擴張，则G是一个free-by-cyclic群。 若F是有限生成群，則稱G是(finitely generated free)-by-cyclic群。 (zh)
- In group theory, especially, in geometric group theory, the class of free-by-cyclic groups have been deeply studied as important examples. A group is said to be free-by-cyclic if it has a free normal subgroup such that the quotient group is cyclic. In other words, is free-by-cyclic if it can be expressed as a group extension of a free group by a cyclic group (NB there are two conventions for 'by'). Usually, we assume is finitely generated and the quotient is an infinite cyclic group. Equivalently, we can define a free-by-cyclic group constructively: if is an automorphism of , the semidirect product is a free-by-cyclic group. (en)
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- Free-by-cyclic group (en)
- Free-by-cyclic群 (zh)
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