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Unformatted text preview: 102 2. BASIC THEORY OF GROUPS Proposition 2.2.32. Let a be an element of ﬁnite order n in a group. For
each positive integer q dividing n, hai has a unique subgroup of order q . Proposition 2.2.33. Let a be an element of ﬁnite order n in a group. For
each nonzero integer s , as has order n=g:c:d:.n; s/.
Example 2.2.34. The group ˚.2n / has order '.2n / D 2n 1 . ˚.2/ has
order 1, and ˚.4/ has order 2, so these are cyclic groups. However, for n
3, ˚.2n / is not cyclic. In fact, the three elements Œ2n 1 and Œ2n 1 ˙ 1
are distinct and each has order 2. But if ˚.2n / were cyclic, it would have
a unique subgroup of order 2, and hence a unique element of order 2. Exercises 2.2
2.2.1. Verify the assertions made about the subgroups of S3 in Example
2.2.2. Determine the subgroup lattice of the group of symmetries of the
2.2.3. Determine the subgroup lattice of the group of symmetries of the
2.2.4. Let H be the subset of S4 consisting of all 3–cycles, all products of
disjoint 2–cycles, and the identity. The purpose of this exercise is to show
that H is a subgroup of S4 .
(a) Show that fe; .12/.34/; .13/.24/; .14/.23/g is a subgroup of S4 .
(b) Now examine products of two 3–cycles in S4 . Notice that the two
3–cycles have either all three digits in common, or they have two
out of three digits in common. If they have three digits in common, they are either the same or inverses. If they have two digits
in common, then they can be written as .a1 a2 a3 / and .a1 a2 a4 /,
or as .a1 a2 a3 / and .a2 a1 a4 /. Show that in all cases the product
is either the identity, another 3–cycle, or a product of two disjoint
(c) Show that the product of a 3–cycle and an element of the form
.ab/.cd / is again a 3–cycle.
(d) Show that H is a subgroup.
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- Fall '08