a What are the possible orders for the elements of a group G

(a) What are the possible orders for the elements of a group G of order 35? (b) What are the possible orders for subgroups of a group G of order 35? (c) Show that every finite group G has 4k elements of order 5, where k is the number of subgroups of order 5 in G.

Solution

(a) The order of any element in a finite group divides the order of the group.

Applying it here, we see that the possible orders of elements are 1,5,7,and 35.

(b) The order of any subgroup of a finite group divides the order of the group.(Lagrange)

Applying it here, the possible orders of subgroups are 1,5,7 and 35.

(c) Let H₁, H₂ , ........H_k be the k subgroups of order 5.

Any two of these have only the identiy element as the common element.

And each non-identity element has order 5.

So each subgroup H_i contains 4 elements of order 5.

So there are totally 4k elements of order 5 in G