# Are there elements of given order in this permutation group?

Let $$S$$ be a group of all permutations of the letters $$U,N,I,V,E,R,S,I,T,Y$$ such that letter $$I$$ is fixed.

Are the statements given below true or false?

a: There exists an element of order 21.

b: There exists an element of order 11.

First of all, I don't know what permutations looks like if elements repeat. But, still I think that if I look at permutation group made of $$U,N,V,E,R,S,T,Y$$ then, it has 8! elements hence, no element of order 11 by Lagrange theorem. Also, order 21 element would be made of two disjoint cycles with 3 elements and 7 elements, respectively, which is impossible here. So, a false as well.

Am I correct?

• The group should be $S_8\times S_2$, and I think you are right. – Hongyi Huang May 28 at 4:50