# Nonabelian group of order 100 every element has order at most 10

I am trying to find an example of a nonabelain group of order 100 in which every element has order at most 10.

I have been trying to use dihedral groups (D_n = {e, f, f^2, … , f^{n-1}, g, fg, f^{2}g, …, f^{n-1}g} ) and have considered the dihedral group D_50. This would have 100 elements so it would have order 100 although I am not entirely sure how to show that each element has order of at most 10.

If anyone has any suggestions on how to prove this or any other suggestions of examples that would be extremely helpful!

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D50D_{50} (the symmetry group of the regular 5050-gon) has an element of order 5050, do that’s no good.
– Arthur
Oct 20 at 20:25

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3 Answers
3

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It seems D5×D5D_5 \times D_5 should work. 100100 elements, non-abelian. Each element in the base groups can have order 1,2,5,101,2,5,10 so the order of an element in the product group, which is the least common multiple of the orders in the base groups, is at most 1010.

That makes sense! Thank you!
– nronnie
Oct 20 at 20:37

I may be missing something but how do you know D_5 X D_5 has 100 elements?
– nronnie
Oct 20 at 20:39

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D5D_5 has 10 elements, so D52{D_5}^2 has 102=10010^2=100 elements.
– Gabriel Burns
Oct 20 at 20:42

2∏i=1D10\prod\limits_{i=1}^2D_{10}

Doesn’t that have 400400 elements? DnD_n has 2n2n elements
– Ross Millikan
Oct 20 at 20:33

DnD_n is order 2n2n not order nn.
– Gabriel Burns
Oct 20 at 20:45

2

There are different conventions for the dihedral groups. One convention makes the number of vertices of the polygon the index, the other the number of elements, @GabrielBurns. Here apparently the latter convention is used. (I prefer the former, but meh.)
– Daniel Fischer♦
Oct 20 at 20:49

D10×D10D_{10}\times D_{10} is an example , another is C10×D10C_{10} \times D_{10}

Moreover, two non-abelian groups with structure (C5×C5):C4(C_5\times C_5):C_4 have no element with order larger then 55.

You mean D5×D5D_{5}\times D_{5}? (see Gabriel Burns answer)
– JeanMarie
Oct 20 at 21:15

GAP, for example, denotes D10D_{10} to be the dihedral-group of order 1010. Otherwise, you can replace D10D_{10} by D5D_5.
– Peter
Oct 20 at 21:16

Sorry, I didn’t know that the 2 notations are coexisting.
– JeanMarie
Oct 20 at 21:26