How do I draw a complete subgroup diagram? The question asks me to give a complete...

How do I draw a complete subgroup diagram? The question asks me to give a complete subgroup diagram for Z₂₅^x

Expert Solution

Let be a group. The relationships among the various subgroups of the group can be illustrated with a subgroup diagram or a subgroup lattice of the group. This is a diagram that includes all the subgroups of the group and connects a subgroup at one level to a subgroup at a higher level with a sequence of line segments if and only if is a proper subgroup of . Although there are many ways to draw such a diagram, the connections between the subgroups must be the same. This diagrams are a very useful way to visuslize a finite group & it's subgroups.

Now the given group in the question is . forms a group under addition modulo n. It is a cyclic group. A cyclic group of finite order has one and only one subgroup of order for every positive divisor of . Divisors of 25 are 1, 5, 25. Subgroups of are (generated by the element , subgroup generated by , & $\left \{ \bar_0 \right \}$ . Now . So the subgroup diagram (or lattice) is :-

Colby Messinger answered 2 years ago

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