Question in Abstract Algebra?

Question

How do I find an example of a non-abelian group that has exactly four elements of order 10.

Answer 1

Actually, Phineas bogg is incorrect - the order of fr in D_10 is 2, not 10: (fr)² = frfr = ffr⁻¹r = e. Indeed, D_10 contains no elements of order 10.

The easiest way to do this is to note that ℤ_10 under addition contains exactly four elements of order 10 - namely, 1, 3, 7, and 9. Of course, ℤ_10 is abelian, but you know that D_20 contains a normal subgroup isomorphic to ℤ_10, and is not abelian. Thus, in D_20, r, r³, r⁷, and r⁹ are elements of order 10, and it is easy to verify that D_20 has no other elements of order 10. So that would be the simplest example.

(N.B. I am using D_n to refer to the dihedral group of order n. Some authors use D_n to refer to the group of symmetries of a regular n-gon, which has order 2n. If this is your book's notation, then in the above, replace D_10 and D_20 with D_5 and D_10, respectively.)

Edit: Yes, D_10 is isomorphic to the semidirect product ℤ₅⋊ℤ₂. Interestingly, the unicode description of ⋊ is "right normal factor semidirect product", even though all the references I've seen that use the symbol use it to refer to the factor on the _left_ being the normal subgroup. I have no idea why that discrepancy exists.

Answer 2

EDIT: Thanks for the correction! Pascal is correct. The flips and rotations of a regular decagon gives such a group. I will leave my old answer here for reference purposes, even thought it is wrong.

Do you happen do know if D_5 is the same as the indirect product of Z2 and Z5? I am guessing that it is, but as you can see my group theory is VERY rusty.


I think the indirect product of Z2 and Z5 would do. An example of this (I believe) is the group of rotations and flips of a regular pentagon. The four elements of order 10 are a flip combined with a non identity rotation. To see that it is nonabelian, notice that a flip and then a 72 degree rotation is different than a 72 degree rotation and then a flip.

Answer 3

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Answer 4

There is only one non-abelian group of order 10.