What is the meaning of $mathbb{Z}_{5}^{+}$ (and $mathbb{Z}_{5}^{*}$) in group theory?
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What does $mathbb{Z}_{5}^{+}$ mean? I know $mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(mathbb{Z}_{5},+)$. I'm confused as to the meaning of this notation?
Edit:
Furthermore, what is the meaning of $mathbb{Z}_{5}^{*}$. The context I saw this in seems to imply this is a group but if it was the same thing as $(mathbb{Z}_{5},*)$ it couldn't be since $(mathbb{Z}_{5},*)$ lacks some inverses.
group-theory finite-groups notation modular-arithmetic
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add a comment |
$begingroup$
What does $mathbb{Z}_{5}^{+}$ mean? I know $mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(mathbb{Z}_{5},+)$. I'm confused as to the meaning of this notation?
Edit:
Furthermore, what is the meaning of $mathbb{Z}_{5}^{*}$. The context I saw this in seems to imply this is a group but if it was the same thing as $(mathbb{Z}_{5},*)$ it couldn't be since $(mathbb{Z}_{5},*)$ lacks some inverses.
group-theory finite-groups notation modular-arithmetic
$endgroup$
$begingroup$
The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
$endgroup$
– M. Nestor
Dec 19 '18 at 3:41
add a comment |
$begingroup$
What does $mathbb{Z}_{5}^{+}$ mean? I know $mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(mathbb{Z}_{5},+)$. I'm confused as to the meaning of this notation?
Edit:
Furthermore, what is the meaning of $mathbb{Z}_{5}^{*}$. The context I saw this in seems to imply this is a group but if it was the same thing as $(mathbb{Z}_{5},*)$ it couldn't be since $(mathbb{Z}_{5},*)$ lacks some inverses.
group-theory finite-groups notation modular-arithmetic
$endgroup$
What does $mathbb{Z}_{5}^{+}$ mean? I know $mathbb{Z}_{5}$ represents the set of integers modulo 5. I would assume this would mean it is the set of integers modulo 5 under addition except that normally this is notated as $(mathbb{Z}_{5},+)$. I'm confused as to the meaning of this notation?
Edit:
Furthermore, what is the meaning of $mathbb{Z}_{5}^{*}$. The context I saw this in seems to imply this is a group but if it was the same thing as $(mathbb{Z}_{5},*)$ it couldn't be since $(mathbb{Z}_{5},*)$ lacks some inverses.
group-theory finite-groups notation modular-arithmetic
group-theory finite-groups notation modular-arithmetic
edited Dec 21 '18 at 17:11
Shaun
9,869113684
9,869113684
asked Dec 19 '18 at 2:21
PolymorphismPrincePolymorphismPrince
1948
1948
$begingroup$
The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
$endgroup$
– M. Nestor
Dec 19 '18 at 3:41
add a comment |
$begingroup$
The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
$endgroup$
– M. Nestor
Dec 19 '18 at 3:41
$begingroup$
The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
$endgroup$
– M. Nestor
Dec 19 '18 at 3:41
$begingroup$
The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
$endgroup$
– M. Nestor
Dec 19 '18 at 3:41
add a comment |
2 Answers
2
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oldest
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For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.
On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.
For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.
$endgroup$
add a comment |
$begingroup$
The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.
[For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]
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Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
$endgroup$
– PolymorphismPrince
Dec 19 '18 at 2:43
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More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
$endgroup$
– Aaron
Dec 19 '18 at 3:27
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.
On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.
For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.
$endgroup$
add a comment |
$begingroup$
For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.
On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.
For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.
$endgroup$
add a comment |
$begingroup$
For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.
On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.
For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.
$endgroup$
For $p$ prime, $mathbb Z_p^*$ is used to denote the multiplicative group of nonzero elements of $mathbb Z_p$.
On the other hand, $mathbb Z_n^×$ denotes the group of units modulo $n$, which has order $varphi (n) $, where $varphi$ is Euler's totient function.
For prime $p$, we have $varphi (p)=p-1$, and indeed $mathbb Z_p^*=mathbb Z_p^×$.
answered Dec 19 '18 at 5:35
Chris CusterChris Custer
14.3k3827
14.3k3827
add a comment |
add a comment |
$begingroup$
The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.
[For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]
$endgroup$
$begingroup$
Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
$endgroup$
– PolymorphismPrince
Dec 19 '18 at 2:43
$begingroup$
More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
$endgroup$
– Aaron
Dec 19 '18 at 3:27
add a comment |
$begingroup$
The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.
[For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]
$endgroup$
$begingroup$
Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
$endgroup$
– PolymorphismPrince
Dec 19 '18 at 2:43
$begingroup$
More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
$endgroup$
– Aaron
Dec 19 '18 at 3:27
add a comment |
$begingroup$
The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.
[For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]
$endgroup$
The notation $mathbb{Z}_n^{times}$ is sometimes used to denote the group of units modulo $n$ with respect to multiplication, so I would presume that $mathbb{Z}_5^+$ denotes the additive group of integers modulo $5$, in order to contrast it with the multiplicative group.
[For what it's worth, I'll point out that $mathbb{Z}_5^{times} cong mathbb{Z}_4^+$.]
answered Dec 19 '18 at 2:39
Clive NewsteadClive Newstead
52k474136
52k474136
$begingroup$
Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
$endgroup$
– PolymorphismPrince
Dec 19 '18 at 2:43
$begingroup$
More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
$endgroup$
– Aaron
Dec 19 '18 at 3:27
add a comment |
$begingroup$
Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
$endgroup$
– PolymorphismPrince
Dec 19 '18 at 2:43
$begingroup$
More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
$endgroup$
– Aaron
Dec 19 '18 at 3:27
$begingroup$
Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
$endgroup$
– PolymorphismPrince
Dec 19 '18 at 2:43
$begingroup$
Thanks for your answer! Just prior to receiving it I edited my question slightly. I'd be very grateful if you could address the last little bit.
$endgroup$
– PolymorphismPrince
Dec 19 '18 at 2:43
$begingroup$
More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
$endgroup$
– Aaron
Dec 19 '18 at 3:27
$begingroup$
More generally, the notation $R^{times}$ is often used to denote the group of units in an arbitrary ring $R$.
$endgroup$
– Aaron
Dec 19 '18 at 3:27
add a comment |
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$begingroup$
The asterick denotes the multiplicative group, not including elements without multiplicative inverses. In the case of $mathbb Z_5^*$ you need only subtract zero but in general $mathbb Z_n^*$ only contains integers relatively prime to $n$.
$endgroup$
– M. Nestor
Dec 19 '18 at 3:41