Let's determine most of the groups with 8 elements. Notice that Lagrange's theorem says that if g G and G has order 8, then o(g) = 1, 2, 4 or 8.

(a) Let G be a group with an element of order 8. State and show what known group is isomorphic to G.

(b) Let G be a group with an element of order 4, and with no element having order larger than 4. Let g have order 4 and let H = . Suppose there is an element in G not in

H that has order strictly less than 4. Show that G is isomorphic to either Z4 Z2 or D4.

(c) Now suppose that G is a group where every element has order less than 4. Show that G is isomorphic to Z2 Z2 Z2. (hint: first argue that G is abelian. Then pick a set

of 3 elements with a special property.)

(There is one missing case from the above. In part b. we assumed that there is an element outside of with order less than 4. The missing case is if there is no such element outside of . That one is a little harder. FYI. There is no question in this part. Just info.)