Recall the following: theorem: (lagrange's theorem) let g be a finite group and hcg. if h is a subgroup of g, then h| divides gl. the purpose of this exercise is to show that the converse of lagrange's theorem is not true in general. recall that al is not abelian, since (1 2), (23) e a4, but (1 2)(23) + (23)(1 2). (a) count the number of elements of order 3 in al. (b) for sake of contradiction, suppose there exists a subgroup hcaof order 6. i. explain why h must be normal in a4. ii. leto e ad have order 3. suppose that o h. arrive at a contradiction. iii. conclude that we must have o eh. arrive at another contradiction. (c) conclude that, although 6 divides the order of a4, as contains no subgroup of order 6.

How can you construct perpendicular lines and prove theorems about perpendicular lines

Given the expression (7 ? 4i) ? (2 + 6i), perform the indicated operation and write the answer in the form a + bi.

Consider the following geometry problems in 3-space enter t or f depending on whether the statement is true or false. (you must enter t or f -- true and false will not work.) f equation editorequation editor 1. a plane and a line either intersect or are parallel t equation editorequation editor 2. two planes parallel to a third plane are parallel f equation editorequation editor 3. two lines either intersect or are parallel f equation editorequation editor 4. two planes orthogonal to a third plane are parallel f equation editorequation editor 5. two planes orthogonal to a line are parallel f equation editorequation editor 6. two planes either intersect or are parallel f equation editorequation editor 7. two lines orthogonal to a third line are parallel t equation editorequation editor 8. two lines parallel to a third line are parallel t equation editorequation editor 9. two planes parallel to a line are parallel f equation editorequation editor 10. two lines parallel to a plane are parallel f equation editorequation editor 11. two lines orthogonal to a plane are parallel