Given an example of three different groups with 8 elements. Why are the groups d

Question

Given an example of three different groups with 8 elements. Why are the groups different?

My related questions are:

1. How to prove that groups are different? My textbook shows this by showing the groups contains different amount of subgroups. Is this the only way to show groups are different?

2. The textbook says there are only 5 different groups of order 8. I know (1) symmetry of square (2) Z8 under addition (3) Quaternion group Q8 (4) U15 group of units of Z15 (since it contains 1 2 4 7 8 11 13 14) which are 8 elements. Are there anymore groups of 8 elements?

Explanation / Answer

The groups Z8, Z4 × Z2, and Z2 × Z2 × Z2 have 8 elements.

Let’s show they are all different. To show their difference we’ll look at the subgroups they have.

Z8 has only one subgroup with 2 elements, namely {0, 4}, while Z4 ×Z2 has 3 subgroups with 2 elements: {(0, 0),(2, 0)}, {(0, 0),(0, 1)}, and {(0, 0),(2, 1)}.

On the other hand, Z2 × Z2 × Z2 has 7 subgroups with 2 elements: {(0, 0, 0),(1, 0, 0)}, {(0, 0, 0),(1, 0, 1)}, {(0, 0, 0),(1, 1, 0)}, {(0, 0, 0),(1, 1, 1)}, {(0, 0, 0),(0, 1, 0)}, {(0, 0, 0),(0, 1, 1)}, {(0, 0, 0),(0, 0, 1)}.

Since all three groups have a different set of subgroups of order 2, they can’t be the same group.

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