Hi! I love your video.. but there's one part of the video which made me confused on how you defined a subgroup. Hence, my question is, is it necessary that the cyclic group contains only one generator? Thanks in advance☺️
Hi! Yes- a group is cyclic if the powers of *one* of its elements comprise *all* elements of the group. However, note that (1) generators for a given cyclic group are usually not unique; for example, 1 is a generator for the group of integers under addition, but -1 is also a generator for that group! And (2) we sometimes use the word “generator” to refer to the (multiple) building blocks of a non-cyclic group... but the word “cyclic” always means “I can find a single element that generates the whole group.”
Cyclic? More like "Sick, these videos are lit!" 🔥🔥🔥
Hi! I love your video.. but there's one part of the video which made me confused on how you defined a subgroup. Hence, my question is, is it necessary that the cyclic group contains only one generator? Thanks in advance☺️
Hi! Yes- a group is cyclic if the powers of *one* of its elements comprise *all* elements of the group. However, note that (1) generators for a given cyclic group are usually not unique; for example, 1 is a generator for the group of integers under addition, but -1 is also a generator for that group! And (2) we sometimes use the word “generator” to refer to the (multiple) building blocks of a non-cyclic group... but the word “cyclic” always means “I can find a single element that generates the whole group.”
what about groups that are generated by two elements. are they significant? is there a special name for them.