Let be a group, a finite subgroup, and suppose . Prove that if and only if .
We have .
Suppose . Then . By a previous theorem, .
Let be a group and a finite subgroup of . Show that if and only if . Deduce that .
It suffices to show that for all , implies . Let . The mapping is a bijection , so that . Since is finite, .
The last statement follows trivially.
Let be a group and let .
Let be a group, and let be normal. Prove that if , then is normal.
Let . Then using lemma 1 to this previous exercise, we have . Thus is normal in .
Let be a group. Prove that the join of any nonempty collection of normal subgroups of is normal.
Lemma: Let be a group, , and a nonempty collection of subgroups of . Then . Proof: Let . Then for some . Thus there exists with , so that . Hence . Let . Then there exists with , so that for some . But , so that .
Recall that the join of is . Now let ; we saw in a previous exercise that . Hence is normal in .
Let be a group.
Let be given by the presentation . Note that every subgroup of is normal, and let . Consider .
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, , . | 4 | |
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Let and .
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Let and let .
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, , and . | 4 | |
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