I couldn't find the words to summarize my question perfectly in the title so I will clarify my question here.
Say we have a group G in which every element can be written in the form [itex]g_1^{e_1} g_2^{e_2}...g_n^{e_n}, 0 ≤ e_i < |g_i| [/itex].
Suppose that there exists a different set [itex] g_1', g_2', ..., g_n' [/itex] that generates G in the same way: [itex]g = g_1'^{e'_1} g_2'^{e'_2}...g_n'^{e'_n}, 0 ≤ e'_i ≤ |g'_i| [/itex] where [itex] |g'_i| = |g_i| [/itex]. (As a concrete example, think of Dn, in which every element can be written as [itex]s^i r^j, 0 ≤ i < 2, 0 ≤ j < n [/itex] and r can be any rotation of order n, and s can be any reflection).
Then suppose [itex] \phi: G → G [/itex] is an automorphism. Then [itex] \phi [/itex] is completely determined by its effect on the generators of G. But are we guaranteed that switching [itex]g_i, g'_i[/itex] always guarantees an automorphism, and that all values of [itex]\phi [/itex] for the generators are independent?
For example, in Dn, can I define [itex]\phi(r)[/itex] to be any rotation with order n, and define [itex]\phi(s)[/itex] to be any reflection? Does the choice of [itex]\phi(s)[/itex] depend on [itex]\phi(r)[/itex]?
Say we have a group G in which every element can be written in the form [itex]g_1^{e_1} g_2^{e_2}...g_n^{e_n}, 0 ≤ e_i < |g_i| [/itex].
Suppose that there exists a different set [itex] g_1', g_2', ..., g_n' [/itex] that generates G in the same way: [itex]g = g_1'^{e'_1} g_2'^{e'_2}...g_n'^{e'_n}, 0 ≤ e'_i ≤ |g'_i| [/itex] where [itex] |g'_i| = |g_i| [/itex]. (As a concrete example, think of Dn, in which every element can be written as [itex]s^i r^j, 0 ≤ i < 2, 0 ≤ j < n [/itex] and r can be any rotation of order n, and s can be any reflection).
Then suppose [itex] \phi: G → G [/itex] is an automorphism. Then [itex] \phi [/itex] is completely determined by its effect on the generators of G. But are we guaranteed that switching [itex]g_i, g'_i[/itex] always guarantees an automorphism, and that all values of [itex]\phi [/itex] for the generators are independent?
For example, in Dn, can I define [itex]\phi(r)[/itex] to be any rotation with order n, and define [itex]\phi(s)[/itex] to be any reflection? Does the choice of [itex]\phi(s)[/itex] depend on [itex]\phi(r)[/itex]?
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