Why Read This Book? - Index of

278 Chapter 8 Groups
One interesting characteristic of the conjugate of a subgroup is that it is also a
subgroup of G.
EXERCISE 8.5.10 Let H be a subgroup of a group G and fix g ∈ G. Then g −1 Hg
is a subgroup of G.
Given two subgroups H1 and H2, to say that H2 is a conjugate of H1 therefore
means that there exists some g ∈ G such that H2 = g −1 H1g.
EXERCISE 8.5.11 Let G be a group, and let H1 and H2 be subgroups of G.
Define H2 ≡ H1, provided H2 is a conjugate of H1. Show that ≡ is an equivalence
relation on the family of all subgroups of G.
EXERCISE 8.5.12 For H ={(1), (12), (34), (12)(34)}, which is a subgroup of S4,
determine the conjugate subgroup (13) −1 H(13).
For another example of a conjugate subgroup, consider D8 as a subgroup of
S4 and let g = (12). For a given δ ∈ D8, the expression g −1 δg can be thought of in
the following way. Since g acts first, it switches the numbers in positions 1 and 2
on the square (an illegal move in D8). Then δ does a rotation and/or flip on this
newly labeled square. Finally, g −1 switches again the numbers in positions 1 and
2 on the square. For example,
g −1 ρg = (12)(1234)(12) = (1342) /∈ D8
(8.72)
Transforming all the elements of D8 in this way creates another subgroup of S4 that
you might need to play around with to understand. It turns out that g −1 D8g as an
algebraic structure is just like D8, except that the rigid square we described before
will not work as a way to visualize it. Instead, picture the numbers {1, 2, 3, 4} being
pushed from corner to corner by g −1 ρg and g −1 φg according to Figure 8.2. The
2
2
3 1
3
1
4
4
g 21 �g
Figure 8.2 Effects of g −1 ρg and g −1 φg on the square.
2
2
3 1
3
1
4
4
g 21�g